Finite-element plasticity and metalforming analysis
Finite-element plasticity and metalforming analysis
G.W. ROWE Professor of Mechanical Engineering University of Birmingham
C.E.N. STURGESS Jaguar Professor of Automotive University of Birmingham
Engineering
P. HARTLEY Senior Lecturer, Department of Manufacturing Engineering University of Birmingham
I. PILLINGER Senior Computer Officer, Department of Mechanical Engineering University of Birmingham
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C A M B R I D G E UNIVERSITY PRESS Cambridge New York Port Chester Melbourne Sydney
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www. Cambridge. org Information on this title: www.cambridge.org/9780521383622 © Cambridge University Press 1991 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1991 This digitally printed first paperback version 2005 A catalogue recordfor this publication is available from the British Library ISBN-13 978-0-521-38362-2 hardback ISBN-10 0-521-38362-5 hardback ISBN-13 978-0-521-01731-2 paperback ISBN-10 0-521-01731-9 paperback
Contents
Preface Acknowledgements Nomenclature
xi xiii xv
1 1.1 1.2 1.3 1.4 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 1.5.6 1.5.7
General introduction to the finite-element method Introduction Earlier theoretical methods for metalforming analysis Basic finite-element approach General procedure for structural finite-element analysis Simple application of elastic FE analysis: a tensile test bar Discretisation Interpolation Stiffness matrices of the elements Assembly of element stiffness matrices Boundary conditions Numerical solution for the displacements Strains and stresses in the elements References
1 1 3 4 5 7 7 7 7 10 10 10 12 12
2 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7
Basic formulation for elastic deformation Types of elements Linear elements Plane-strain triangular elements Linear quadrilateral elements Higher-order (quadratic) elements Three-dimensional elements Size of elements Shape and configuration of elements: aspect ratio
14 14 15 16 16 16 17 17 18
2.2 2.3
Continuity and equilibrium Interpolation functions
18 19
Contents
2.3.1 Polynomials 2.3.2 Convergence
19 19
2.4
Displacement vector u and matrix [B]
20
2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.6
The [D] matrix of elastic constants General expression Plane elastic stress Plane elastic strain Axial symmetry Formulation of the element stiffness matrix [K{]
22 22 23 24 24 25
2.7 2.7.1 2.7.2
Formulation of the global stiffness matrix [K] Assembly of element matrices Properties of [K]
26 26 27
2.8 2.8.1 2.8.2 2.8.3
General solution methods for the matrix equations Direct solution methods Indirect solution methods Iteration to improve a direct solution
31 31 33 35
2.9
Boundary conditions
35
2.10 2.10.1 2.10.2 2.10.3
Variational methods Variational method of solution in continuum theory Approximate solutions by the Rayleigh-Ritz method Weighted residuals
36 36 37 37
2.10.4 Galerkin's method 2.11
38
The finite-difference method
38
References
39
3
Small-deformation elastic-plastic analysis
40
3.1
Introduction
40
3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7 3.2.8
Elements of plasticity theory 40 Yielding 40 Deviatoric and generalised stresses 42 Constancy of volume in plastic deformation 42 Decomposition of incremental strain 43 Generalised plastic strain 43 The relationship of stress to strain increment: the Prandtl-Reuss equations 44 Elastic-plastic constitutive relationship 45 Strain hardening in FE solutions 45
3.2.9 Force/displacement relationship: the stiffness matrix [K{] 3.3
46
Example analysis using the small-deformation formulation
46
References
48
Finite-element plasticity on microcomputers
49
4.1
Microcomputers in engineering
49
4.2
Non-linear plasticity demonstration programs on a microcomputer using BASIC
50
4
Contents
4.2.1 Introduction 4.2.2 Structure of the finite-element program in BASIC 4.2.3 Simple application and comparison to mainframe results 4.3
A 'large' FORTRAN-based system for non-linear
finite-element plasticity 4.3.1 Hardware selection 4.3.2 Program transfer 4.3.3 Applications of the non-linear system 4.3.3.1 Axi-symmetric upsetting 4.3.3.2 Cold heading 4.3.4 Overcoming the FORTRAN compiler limitations 4.3.5 Introducing an improved FORTRAN compiler and the 8087 maths co-processor 4.3.5.1 System improvements 4.3.5.2 Analysis of upsetting and heading with refined meshes 4.4
50 51 52 56 56 58 58 58 60 61 62 62 63
Summary
64
References
65
5
Finite-strain formulation for metalforming analysis
66
5.1
Introduction
66
5.2 5.2.1 5.2.2 5.2.3 5.2.4
Governing rate equations Rate of potential energy for updated-Lagrangian increment Minimisation of rate of potential energy Incorporation of strain rate Importance of correct choice of stress rate
67 67 68 68 69
5.3 5.3.1 5.3.2 5.3.3
Governing incremental equations Modification of rate expression Effect of rotation LCR expression for strain increment
70 70 71 72
5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5
Elastic-plastic formulation Yield criterion Elastic-plastic flow rule Elastic-plastic constitutive relationship Effect of plastic incompressibility Element-dilatation technique
73 73 74 74 74 75
5.5 Element expressions 5.5.1 Interpolation of nodal displacement 5.5.2 Incremental element-stiffness equations References
77 77 78 78
6
Implementation of the finite-strain formulation
80
6.1
Introduction
80
6.2
Performing an FE analysis - an overview
80
6.3 Pre-processing 6.3.1 Mesh generation
82 82
Contents
6.3.2 6.3.3 6.3.4 6.3.5
Boundary conditions Material properties Deformation Initial state parameters
6.4 FE Calculation 6.4.1 Input of data 6.4.2 Displacement of surface nodes 6.4.3 Assembly and solution of the stiffness equations 6.4.3.1 Incorporation of frictional restraint 6.4.3.2 Solution techniques 6.4.3.3 Yield-transition increments 6.4.4 Updating of workpiece parameters 6.4.4.1 Strain components and rotational values 6.4.4 .2 Deviatonc stress 6.4.4 .3 Hydrostatic stress 6.4.4 .4 External forces 6.4.4 .5 Strain rate 6.4.4 .6 Temperature 6.4.5 Output of results 6.5 Post-processing 6.6 Special techniques 6.6.1 Processes involving severe deformation 6.6.2 Steady-state processes 6.6.3 Three-dimensional analyses References
83 85 85 87 87 87 88 90 90 92 95 95 95 96 99 100 100 101 107 108 108 108 111 112 114
7
Practical applications
116
7.1
Introduction
116 117 117 122 122 125 131 135 137 145 149 149 153 153 155 159 159 163 167
7.2 Forging 7.2.1 Simple upsetting 7.2.2 Upset forging 7.2.3 Heading 7.2.4 Plane-strain side-pressing 7.2.5 Rim-disc forging 7.2.6 Extrusion-forging 7.2.7 'H'-section forging 7.2.8 Forging of a connecting rod 7.3 7.3.1
Extrusion Forward extrusion
7.4 7.4.1 7.4.2
Rolling Strip rolling Slab rolling
7.5 7.5.1 7.5.2
Multi-stage processes - forging sequence design Automobile spigot Short hollow tube (gudgeon pin) References
Contents
8
Future developments
169
8.1
Introduction
169
8.2
Developments in software
170
8.3
Advances in modelling
170
8.4
Material properties
172
8.5
Post-processing
173
8.6
Expert systems
174
8.7
Hardware
174
8.8
Applications in the future
175
References
176
Appendices
178
1
Derivation of small-strain [B] matrix for 2-D triangular element
178
2
Derivation of elastic [D] matrix
181
3
Derivation of elastic-plastic [D] matrix
184
4
Derivation of small-strain stiffness matrix [K] for plane-stress triangular element
188
5
Solution of stiffness equations by Gaussian elimination and back-substitution
191
6
Imposition of boundary conditions
196
7
Relationship between elastic moduli E, G and K
202
8
Vectors and tensors
205
9
Stress in a deforming body
209
10
Stress rates
216
11
Listing of BASIC program for small-deformation elastic-plastic FE analysis
218
Bibliography
246
Index
294
Preface
More and more sectors of the metalforming industry are beginning to recognise the benefits that finite-element analysis of metal-deformation processes can bring in reducing the lead time and development costs associated with the manufacture of new components. Finite-element analyses of non-linear problems, such as metal deformation, require powerful computing facilities and large amounts of computer running time but advances in computer technology and the falling price of hardware are bringing these techniques within the reach of even the most modest R & D department. Finite-element programs specially designed for the analysis of metalforming processes (such as the program EPFEP3*, which is used for the majority of examples in this monograph) are now commercially available. As a result, there is an urgent need for a text that explains the principles underlying finite-element metalforming analysis to the people who are starting to use these techniques industrially, and to those undertaking the undergraduate and postgraduate engineering courses in the subject that industry is beginning to demand. One of the aims of this monograph is to fulfil that need. The first chapters provide an introduction to the application of finiteelement analysis to metalforming problems, starting from the basic ideas of the finite-element method, and developing these ideas firstly to the study of linear elastic deformation and then to the examination of non-linear elastic-plastic processes involving small amounts of deformation. No previous knowledge of finite elements is required, although the reader is assumed to be familiar with the use of matrices. Chapter 4 describes a program, written in BASIC for a PC-type desktop computer, that uses the simple, small-deformation formulation. Although restricted to two-dimensional metal flow, low levels of deformation and a small ::
Copyright © 1987 University of Birmingham All Rights Reserved.
xii
Preface
number of elements, this program demonstrates the use of finite-element analysis in metalforming and is particularly useful for tuition purposes. A full listing is given in Appendix 11. Chapter 4 also discusses implementations of the smalldeformation theory on more powerful microcomputers that have larger memories and access to a FORTRAN compiler. These programs are capable of more detailed analysis, and can be used for the study of many simple two-dimensional processes. The application of finite-element techniques to metalforming is still the subject of much research, and the second aim of this monograph is to present the current state of knowledge to those new to the field. Chapter 5 therefore examines in detail the theoretical aspects of a finite-element model of large-deformation plasticity, and Chapter 6 explains how this theory can be implemented in a practical program for the study of complex metalforming problems. Chapter 7 describes a wide variety of practical applications of the finite-element programs. Although, where appropriate, we mention some of the other finite-element techniques that have been applied to plasticity problems, this monograph is written from the standpoint of an elastic-plastic approach to metal deformation. We are firmly of the opinion that the various simplifications that are sometimes made are a false economy and a serious hindrance to the further development of the finite-element study of metal flow. The last chapter of this monograph identifies the main areas where we believe that future developments will take place, or at least those areas in which work needs to be done. Unfortunately, the gaps in the present state of knowledge of metalforming processes and the deficiencies in the available models of metal deformation are all too obvious. We hope that this monograph will provide a stimulus to others to attempt to remedy these failings. G.W.R. C.E.N.S. P.H. I.P.
Acknowledgements
Many people have been associated over the years with the Finite-element Plasticity Group at the University of Birmingham and have contributed, directly or indirectly, to the development of large-deformation elastic-plastic finite-element techniques. We are particularly indebted to Profs. C. Liu and K. Wang, Drs S.E. Clift, A.A.M. Hussin, A.A.K. Al-Sened, A.J. Eames and J. Salimi and Mr K. Kawamura, whose research work has influenced the contents of this monograph. We should also like to thank Mr S.K. Chanda, Profs. S. Ikeda and K. Kato, as well as the above persons, for providing the numerous examples of finite-element metalforming applications that are to be found in the text. In addition, both undergraduate and postgraduate students have contributed through a variety of short projects. Many of the overseas research students and visiting research fellows have been able to work with us at Birmingham through the generosity of their respective governments and, in some cases, with additional help from the British Council. We are grateful to these bodies for their support. Much of our research work has been carried out with the financial support of the UK Science and Engineering Research Council, and more recently the ACME Directorate, and with computer facilities provided by the Centre for Computing and Computer Science at the University of Birmingham and the University of Manchester Regional Computer Centre. Our contacts with industry have been particularly useful to us and we are grateful for the advice and help we have obtained, especially from Alcan Plate, Austin-Rover, Automotive Products, GKN and High Duty Alloys. We are very grateful to Mr B. Van Bael for checking the manuscript and particularly for wading through the mathematics. It is to his credit if the text emerges free of error. If any mistakes remain, they are entirely our fault, not his. We were very saddened by the death of Professor Rowe during the preparation of this monograph. Much of the work described in the following pages was begun at his instigation and carried out under his guidance. Indeed, that the monograph
xiv
Acknowledgements
came into existence at all is very much due to his energy and his enthusiasm. We owe a great debt to him and feel privileged to have known him and to have worked with him over many years. In completing the book, we have been very conscious of the high standards he set in everything he wrote. We hope that the result does not compare too unfavourably with those standards. We should like to dedicate this work to him. C.E.N.S RH. I.P.
Nomenclature
Use of subscripts and superscripts lower-case subscript in italic type: usually indicates a Cartesian component of a quantity, e.g. xt. If preceded by
a comma, it indicates a derivative with respect to a particular Cartesian coordinate, e.g. Uij = dujdxj. lower-case subscript in bold type:
indicates the quantity associated with a particular element, e.g. L x. upper-case subscript in italic type:
indicates the quantity associated with a particular node of an element or body, e.g. Nj. Greek subscript:
indicates the quantity associated with a particular degree of freedom of the whole workpiece. In the simple one-dimensional examples considered in the early part of this monograph, a Greek subscript therefore denotes the value of a quantity for a particular node of the workpiece, e.g. da. lower-case superscript in italic type:
used in the notation for contravariant and mixed tensors, but only in Appendix 8 (vectors and tensors). superscript in normal type:
is a label, rather than a numerical index, denoting a particular type of subset of a quantity, e.g. ATOest. Some frequently-used superscripts are: T transpose of vector or matrix p plastic portion ' (prime) for tensors - deviatoric component, e.g. &-^ for positional quantities - transformed position, e.g. x\
xvi
Nomenclature for scalars - derivative, e.g. Y Other examples are given in the list of symbols below.
lower-case superscript in parentheses: indicates the value of a quantity for a particular iteration, e.g. Ad (2) . The parentheses are used to avoid confusion with an exponent. Greek superscript: indicates the quantity associated with a particular face of the element under consideration, e.g. Aa. Specific values of bold subscripts are printed using bold numerals but when numbers are substituted for italic subscripts normal type is used. No distinction is made between numbers substituted for lower-case subscripts and those substituted for Greek subscripts. The meaning is made clear by the context.The context will also make clear whether, for example, X2 means the square of the value of X, or the second contravariant component of the vector X. The different types of subscript and superscript may be used together, e.g. fia or q%
Frequently-used qualifying symbols A dot above a quantity denotes the time derivative, e.g. dIm. A bar above a tensor denotes the generalised value, e.g. a.
Representation of vectors and matrices A vector is represented by bold type, a matrix by a symbol enclosed in square brackets. Both may also be notated by specifying the symbol (without bold type or brackets) with algebraic subscripts (and possibly superscripts). This may represent a particular element of the vector or matrix, or may be intended to stand for the whole array, with the indices implicitly varying over all their possible values, e.g. [K] = Kap. When the elements of a vector are written out explicitly, they are enclosed in parentheses. If they are written as a row vector, the elements are separated by commas. The elements of a matrix are simply enclosed in square brackets, without any commas.
List of symbols A 12M
A Ax Aa a
area area of quadrilateral 1234 area of element / area of face a of element coefficient of general polynomial shape function. Also coefficient
Nomenclature
#,atj da da' B [B] [By] Bijlm Bijlm
b bt by
C j + , O~
c c, [D] [De] [Dp] [Di] Dy DijM d d d da
xvii
of 1-D linear displacement function coefficient of linear function for displacement in xt direction coefficient of expression determining /th reference co-ordinate of a point as a quadratic function of local curvilinear co-ordinates X7 infinitesimal area at P at time t infinitesimal area at P' at time t+dt coefficient of yield-stress function matrix that may be used to express the strain at a point within an element in terms of the displacement of its nodes strain/nodal displacement matrix for element / coefficient of LCR strain increment/nodal displacement increment matrix (= Vi (BijIm +BjiIm) ) coefficient of defining matrix for LCR strain increment/nodal displacement increment matrix (= rJSfij ~
(SjmNIik -dkmNu)l4) coefficient of general polynomial shape function. Also coefficient of 1-D linear displacement function coefficient of general linear shape function coefficient of linear function for displacement in x{ direction. Also coefficient of expression determining /th reference co-ordinate of a point as a quadratic function of local curvilinear co-ordinate X1 coefficients of expression determining rate of flow of heat at element centroid as a function of the difference in temperature between the centroid and the point on the yth local curvilinear axis with local co-ordinate +1 or - 1 coefficient of general polynomial shape function. Also thermal capacity per unit volume coefficient of linear combination of functions that form 0 6 x 6 stress/strain constitutive matrix such that a = [D]e elastic part of elastic-plastic constitutive matrix plastic part of elastic-plastic constitutive matrix constitutive matrix for element / entry in row /, column / of [D] coefficient of constitutive tensor relating Jaumann rate of stress T]J to component e*/ of stain-rate tensor displacement. Also thickness of friction layer global nodal displacement vector approximate solution of equations derived by incomplete Choleski conjugate-gradient method (= [£/*]d) component a of the nodal displacement vector. For 1-D problems, this is the displacement of node a
xviii d*a d/ di d} djj Ad Ada
Adc
Ad^ Adn dd E E{ e F / f f f* f f+ fa fa fl f7 fi f/
Nomenclature component a of nodal displacement vector after modification during Gaussian elimination procedure vector of components of displacement of node / vector of components of displacement of nodes of element / vector of components of displacement of node / in rotated axis system displacement of node / of element in x; direction representative displacement increment first estimate of displacement increment during secant-modulus technique, evaluated by solving stiffness equations obtained for initial strain 6° and stress <x° second estimate of displacement increment during secant-modulus technique, evaluated by solving stiffness equations obtained for mid-increment strain em and stress cr™ ith correction to the incremental displacement in initial-stiffness iteration change in dn during an increment of deformation error in estimate of d Young's Modulus of elasticity Young's Modulus for the material of element / small extension of an element functional of state functions in variational method. Also number of faces of an element in contact with a die force global nodal force vector. Also force acting at a point P of a body force acting at a point P' of a body at time t+dt right-hand side of equations derived by incomplete Choleski conjugate-gradient method (= [L*] -1f) force acting at time t+dt on an infinitesimal plane with area da' and a normal that has reference components equal to N\ global force vector corresponding to a global displacement vector of d+Sd component a of the nodal force vector. For 1-D problems, this is the resultant force at node a component a of the nodal force vector after modification during the Gaussian elimination procedure contravariant reference component of f vector of components of force applied to node / vector of components of force applied to nodes of element i vector of components of force applied to node / in rotated axis system
Nomenclature fl /// fia A/
A/// G G/ G' gi gi(x) gi g' H h h(d) hij I
/i / K [K] [K*]
xix
contravariant reference component of f component of force in xt direction applied to node / of element force applied to element / at node a representative force increment resultant force calculated in ith iteration of initial-stiffness technique force in equilibrium with Ad^ in initial-stiffness iteration change in fn during an increment of deformation Rigidity (shear) Modulus of elasticity (= E/2(l + v) ). Also functional of state functions used in variational method basis of 3-D vector space basis of 3-D vector space (dual of G/) stiffness coefficient of element / (= A{EJL^ one of the simple functions of position used to define an approximation to the function of state in variational method basis of 3-D vector space basis of 3-D vector space (dual of g,) initial height of billet final height of billet scalar function of n variables that has a minimum value for the vector solution of the stiffness equations matrix used in the evaluation of rotational values (= qn
(= [L*V [^[t/T 1 ) [^i] Ka/3 K*ap [Ku]
KImJn
stiffness matrix of element i entry in global stiffness matrix entry in global stiffness matrix after modification during Gaussian elimination procedure 2 x 2 (for 2-D) or 3 x 3 (for 3-D) submatrix of coefficients of global stiffness matrix relating components of force at node / to components of displacement at node / coefficient of element stiffness matrix relating component n of incremental displacement of node / to component m of incremental force applied to node /
xx KeimJn K'imjn K^imjn k k*+, k)~ L [L] [L*] Lx l']+, V~ m ma Am N N(x) N't iV/(x) Nu n n n' nt n\ O O' P P' p p(R) P 1, P 2 P 3 pty q
Nomenclature coefficient of element deformation stiffness matrix coefficient of element stress-increment correction matrix coefficient of element dilatation matrix shear yield stress. Also thermal conductivity coefficients of heat transfer between die and face of element with centre on positive or negative ;th local curvilinear axis length lower-triangular matrix derived from [K] during Choleski decomposition approximation to [L] having a pattern of sparseness similar to that of[K\ length of element i distance between centroid of element and face with centre on positive or negative yth local curvilinear axis friction factor. Also exponent in yield-stress function friction factor associated with die in contact with face a of element proportionality factor in modified Prandtl-Reuss equations number of nodes in an FE discretisation. Also number of steps in thermal calculation for an increment of deformation general shape function of position covariant component of n' in the convected co-ordinate system shape function of position relating to the contribution from node / gradient of N7 (xj) (= dJV7 (xj) Idx) number of degrees of freedom in an FE discretisation normal to infinitesimal surface at P at time t. Also normal to boundary surface at P at start of increment normal to infinitesimal surface at P' at time t+dt. Also normal to boundary surface at P' at end of increment covariant components of n in reference co-ordinate system covariant components of n' in reference co-ordinate system origin of reference co-ordinate system origin of convected co-ordinate system location of infinitesimal region of body at time t. Also coefficient used in calculating number of thermal steps location of infinitesimal region of body at time t+dt coefficient of general polynomial shape function. Also coefficient of quadratic function of Am (= 3d-'ijdJij/2) function of residual R in weighted residual method planes of constraint of nodes result of dividing htj by /th estimate of qtj coefficient of quadratic function of Am (= — 3<x- 7- (a?j
Nomenclature qij qtj q\f 8QC
xxi
symmetric part of deformation matrix x'l'j symmetric part of deformation matrix x'tj /th estimate of q^ using Newton's method of solving equation
increase in energy of element in one step of thermal calculation due to conduction of heat into and out of element 8Qd increase in energy of element in one step of thermal calculation due to work of deformation increase in energy of element in one step of thermal calculation 8Qi due to frictional heating at boundaries R residual functional of state functions in weighted residual method [R] orthonormal (rotational) matrix relating global to rotated components of a vector r order of polynomial interpolation function. Also coefficient of quadratic function of Am (= 3(o?j +Aa?j) 2 12) r right-hand side vector of equations obtained by Gauss-Jordan elimination r{ coefficient of expression determining temperature of a point as a quadratic function of local curvilinear co-ordinates XJ rij rotational (orthogonal) part of deformation matrix xllii rij rotational (orthogonal) part of deformation matrix x\j S coefficient in elastic-plastic constitutive relationship (=3/2d-2 [1 + (Y73G)] ) 5 j + , S]~ coefficients generalising element heat-flow expression to external faces s thickness of an element s right-hand side vector of equations obtained by Gaussian elimination S( coefficient of expression determining t e m p e r a t u r e of a point as a quadratic function of local curvilinear co-ordinates Xj ij s contravariant c o m p o n e n t of nominal or Piola-Kirchhoff I stress Sy covariant c o m p o n e n t of nominal o r Piola-Kirchhoff I stress T absolute t e m p e r a t u r e . Also tensile stress acting in single-element example 7* t e m p e r a t u r e at e n d of increment of deformation 1 T t e m p e r a t u r e at start of increment of deformation 7°, Tl temperature at beginning and end of time step At. T° is also temperature at the centroid of an element T j + , T]~ temperature at point on /th local curvilinear axis with local coordinate -hi or - 1 [7] diagonal matrix derived from [K] by Gauss-Jordan elimination
Nomenclature
7^ A 7° AT°e s t t tij' t° 8T dt At U [U] [U*] Ux u{x) u ul ut ux Uij u1^ u^ Autj V V* VI Vt V\ Vij V'ij v
estimate of temperature of centroid at end of /th thermal step contravariant component of a general tensor in co-ordinate system JC change in temperature during an increment of deformation estimated change in temperature during an increment of deformation time contravariant component of a general tensor in co-ordinate system xl initial time change in temperature during one step of thermal calculation infinitesimal time increment time interval for increment of deformation strain energy upper-triangular matrix derived from [K] by Gaussian elimination or Choleski decomposition approximation to [U] having a pattern of sparseness similar to that oi[K] strain energy of element / linear displacement of point from its initial position x in 1-D example vector of displacement of a point or sometimes a typical vector contravariant component of displacement of a point covariant component of displacement of a point; for Cartesian co-ordinate system, displacement of a point in xt direction displacement of a point of element / gradient of component / of displacement in direction Xj rate of deformation matrix in terms of contravariant components ( = d(jt"'') I At = xfi>j) rate of deformation matrix in terms of covariant components ( = d(*; v ) i&t = *;,,) deformation gradient; change in w/; during an increment of deformation (= Ax'ij) volume volume in local element co-ordinate system contravariant component of v in co-ordinate system X covariant component of v in co-ordinate system X volume of element i yth convected component of V /th convected component of v" typical vector
Nomenclature vl V; \l \n vi] v'ij W W{ w X X
Xi
X'1
x x1, x2 x xl
Xi
x°i xj + , x\~ x11 x'i xn xn'j
x'i j
contravariant component of v in co-ordinate system xl covariant component of v in co-ordinate system xl arbitrary infinitesimal vector at P at time t arbitrary infinitesimal vector \l after deformation of P to P' yth reference component of \l yth reference component of v" work done by external forces work done by external forces on element / multiplication factor in Gaussian elimination procedure coefficient used in calculation of number of thermal steps co-ordinate i of a point P in rotated, convected or transformed axis system (specifically, contravariant component with respect to basis G/) co-ordinate / of a point P in rotated, convected or transformed axis system (specifically, covariant component with respect to basis G<) co-ordinate i of a point P' in rotated, convected or transformed axis system (specifically, contravariant component with respect to basis G/) linear co-ordinate lower and upper limits of integration in variational expression position vector of a point co-ordinate i of a point P in initial reference (usually Cartesian) axis system (specifically, contravariant component with respect to basis gi) co-ordinate i of a point P in initial reference (usually Cartesian) axis system (specifically, covariant component with respect to basis g1) /th reference co-ordinate of centroid of an element /th reference co-ordinate of point on yth local curvilinear axis with local co-ordinate + 1 or —1 co-ordinate / of point P' in initial reference axis system (specifically, contravariant component with respect to basis gt) co-ordinate i of point P' in initial reference axis system (specifically, covariant component with respect to basis gl) co-ordinate / of node / of an element contravariant co-ordinate transformation matrix defining deformation of infinitesimal region at point P at time t into infinitesimal region at point P' at time t+dt (= dxfi ldxj) covariant co-ordinate transformation matrix defining deformation of infinitesimal region at point P at time t into infinitesimal region
Nomenclature
x*
lyJ
Ax'ij Y V Z a /3 jij A 8 8 () Sij e e^ e° 6C 6m ep eOp 6P e 6/ €[ €ij eij €ij e'ij de p defy de p ;
at point P' at time t+dt (= dx\ IdXj) transformation matrix defining a rigid-body rotation; rotational part of *"'•'•(= r ij) deformation gradient; change in xfitj during an increment of deformation ( = Auifj) yield stress in simple tension, a function of strain, strain rate and temperature rate of change of Y with respect to plastic strain coefficient used in calculation of number of thermal steps angle. Also proportion of work of deformation converted into heat angle engineering shear strain in x&j plane ( = 2eijy i±j) change in the value of a quantity during an increment of deformation increment of a quantity arbitrary variation of function enclosed in parentheses Kronecker delta ( = 1 , i=j; = 0 , i±j). Also 8), & etc strain normal strain in AB direction representative strain at start of increment representative strain at e n d of increment calculated by secantm o d u l u s technique ( = 6 ° + A ec ) representative strain during an increment calculated by secantm o d u l u s technique ( = eo+ViAe*) accumulated generalised plastic strain ( = Jde p ) plastic strain at start of plastic part of increment of deformation rate of change of plastic strain with respect to time vector of 3 normal components of strain and 3 engineering shear components of strain principal component of strain strain in element / normal component of strain in xt direction if i=j; shear component of strain in xtXj plane if i+j contravariant component of strain-rate tensor (= Vi{iilti + ujjl) ) covariant component of strain-rate tensor, equivalent to e^' for Cartesian (orthogonal) co-ordinate system (= Vi(iiitj + iijj) ) deviatoric component of strain (= e^ - 8i}€kk/3) increment in generalised plastic strain (= (2de?)de?)/3) 2) elastic part of incremental strain component plastic part of incremental strain component
Nomenclature Ae a A ec A ep Aew Ae^ r] K dA AA v a aAB <j° ac a^ a"1 a cr (Ti ax (Tij
a'ij a°i'j a'ij a'ij A a^l) A a\j Aaei'j
xxv
strain increment corresponding to a displacement of Ad a , the first estimate of displacement, in secant-modulus technique strain increment corresponding to a corrected displacement of A dc, in secant-modulus technique change in plastic strain during an increment of deformation strain increment calculated from Ad^ in initial-stiffness iteration component of LCR increment of strain (specifically, change in strain during plastic part of an increment) coefficient of viscosity Bulk Modulus of elasticity ( = E/3(l-2i/) ) proportionality factor in Prandtl-Reuss elastic-plastic flow equations proportionality factor in incremental form of Prandtl-Reuss elastic-plastic flow equations Poisson's Ratio Cauchy or True stress normal stress acting in AB direction representative stress at start of increment representative stress at e n d of increment corresponding t o a strain of 6C hydrostatic c o m p o n e n t of stress ( = o-kk/3) representative stress during an increment corresponding to a strain of 6 m generalised stress ( = (3a[^a'^/2) 2 ) vector of 6 unique c o m p o n e n t s of stress (since atj = cr;/) principal c o m p o n e n t of stress stress in element / normal component of stress in xt direction if /=/; shear stress resulting from force in xj direction acting on surface with normal in xt direction if i+] deviatoric component of stress (= a^ — 8^) deviatoric stress at beginning of plastic part of deformation increment deviatoric stress half-way through hypothetical elastic step of deformation increment (= afj +VzAaCij) deviatoric stress half-way through plastic part of step (= a°i'j +V2Aa'u) stress increment calculated from A e (/) in initial-stiffness iteration change in deviatoric stress during plastic part of deformation increment (= AaYj-2G Aka'ij) hypothetical elastic change in stress; change in stress that would
xxvi
T T*7 ¥ J' T li
i'ij' T]J cf)(x) 4>', 4>" <£(x) A (/>(*/) A >, < > II { } * det ( )
Nomenclature result if plastic part of deformation increment took place elastically (= 2GAeij) maximum permitted error in calculated incremental temperature change contravariant component of Kirchhoff or Piola-Kirchhoff II stress time derivative of rij rotationally-invariant Kirchhoff stress; contravariant component of Kirchhoff stress at a point in a co-ordinate system that rotates with the deforming infinitesimal region at that point contravariant component of Jaumann rate of Kirchhoff stress covariant component of Jaumann rate of Kirchhoff stress function of state in a physical system to be determined by variational method first and second derivatives of cf) with respect to position approximation to
1
General introduction to the finite-element method
1.1
INTRODUCTION
Metalforming is an ancient art and was the subject of closely-guarded secrets in antiquity. In many respects the old craft traditions have been retained until the present time, even incorporating empirical rules and practices in automated production lines. Such techniques have been successful when applied with skill, and when finely adjusted for specific purposes. Unfortunately, serious problems arise in commissioning a new production line or when a change is made from one well-known material to another whose characteristics are less familiar. The current trend towards adaptive computer control and flexible manufacturing systems calls for more precise definition and understanding of the processes, while at the same time offering the possibility of much better control of product dimensions and quality. This is especially important in the advanced aerospace and other industries, where the advantages of new materials have to be fully exploited as quickly as possible. Alloys that have been specifically designed to withstand high temperatures provide one example where the tools are stressed to their limits and the alloys themselves may be deformable only within narrow ranges of temperature and strain-rate. Practical tests to determine the best tool shapes and forming conditions can be very expensive and wasteful of scarce tool and workpiece material. A single turbine blade may cost several thousand pounds, so any reduction in prototype manufacturing costs is clearly valuable. Even the common forgings used in large numbers for automobiles may be subject to cracking during production, or be weakened by high residual stresses, if the preforms are incorrectly designed. It is therefore important to provide guidance from theoretical models which can easily be modified. Until quite recently the only available theories of metalforming operations were based on simple equilibrium, ignoring internal distor-
General introduction to the finite-element method
tion, or else made gross assumptions about the properties of the materials [1.1]. Both slip-line field theory and upper-bound theory, in the latter category, have been advanced by the introduction of computer methods, but there is no doubt that the greatest improvement in detail and accuracy has come from the application of finite-element (FE) analysis. Originally the FE method of stress analysis was restricted to linear elastic deformation, in which form it has many uses in structural design. Many books have been written on this subject [1.2, 1.3, 1.4] but the main concern of this monograph will be with plastic deformation. It is now possible to take into account the practical non-linearities in shape and material properties, and to produce accurate predictions of stress, strain, strainrate and temperature distributions throughout a plastically-deforming workpiece. The analyses can even be used to determine whether the alloy is likely to fail by tensile cracking or catastrophic shear during processing. Information about the surface pressures, tractions and overall load is also made available and can be used as input conditions for elastic FE analysis of tool designs. The results of such comprehensive analysis provide a sound basis for development of control strategy in metalforming. A complete processing route, including the effects of interstage annealing, can thus be examined theoretically to determine optimum preform shapes from the point of view of product quality or economic manufacture. The information provided is fully detailed and can be incorporated in Computer-Aided Design/ Computer-Aided Manufacturing (CAD/CAM) sequences for tool production, or as a basis for analyses of fatigue resistance or fracture liability during normal use of the product. This monograph develops the analytical procedures from an elementary starting point, needing no prior knowledge of FE methods. The basis of the programs is fully explained, without requiring the reader actually to write programs personally unless so desired, and examples of their use in practical metalforming situations are given. Most FE analysis up to the present date has required very large main-frame computers, but the rapid advances in microcomputers during the last decade are changing the pattern, especially in civil engineering [1.5]. Chapter 4 shows that, at the expense of greater computing time but not necessarily of overall turn-round time, FE analyses for both elastic and plastic problems can be carried out on the latest large-memory micros. There is a very great saving in capital cost, and a further industrial advantage of immediate in-house availability of the results. As in all computer packages however, there is danger in using the programs without a background understanding of the principles involved and the limitations imposed by the assumptions and structure of the analysis. It is intended that this monograph will provide a basis for confidence in using FE plasticity analysis in industrial metalforming, as well as being a self-contained
1.2 Earlier theoretical methods
text-book for specialist graduate or undergraduate courses in forging, extrusion, rolling and related processes. It also provides a background for more general non-linear FE analysis, as applied, for example, to polymeric materials.
1.2
EARLIER THEORETICAL METHODS FOR METALFORMING ANALYSIS
The earliest theoretical models were based on the concept that the energy expended in forging could be attributed to the internal deformation, including both shape change and internal distortion, and the external resistance at the tool faces [1.6]. Empirical formulae were developed containing terms dependent upon the flow stress of the alloy and upon the coefficient of friction. These allowed fair predictions of overall forces to be made. Further detail of pressure distributions can be obtained by analysing the local stress equilibrium. This also predicts the forces, and the results are helpful when designing tools and choosing tool materials. Although the method is intended for simple shapes, useful approximations can often be made for real problems, especially in rolling [1.7]. Slip-line theory gives much more detailed information about the flow patterns, as well as the stress distributions and overall forces. This technique has been applied to forging, forward and backward extrusion, drawing and other processes, but it makes very restrictive assumptions about the material properties [1.8]. Only the average yield stress is included as a variable, and elastic deformation is ignored. The rigorous theory is, moreover, confined to plane-strain deformation. Originally the method was very slow, requiring manual calculation and drawings, but it can now be used on a computer [1.9]. The only serious competitors to the FE Method (FEM) among the currently available theories are Finite-Difference Methods, Boundary-Element Methods and the Upper-Bound (UB) technique [1.10]. In its computerised form, the latter offers a rapid and relatively simple way of calculating the major force requirements [1.11]. It does however require experience in choosing the appropriate basic units of the shear field. Computer packages are now available for well-established configurations or assemblies of proven elements, applicable to plane strain or axial symmetry. More elaborate versions are becoming available [1.12], some of which include temperature and strain-rate effects, but all suffer from the necessity to make assumptions about the internal deformation patterns. They are consequently unlikely to be reliable for prediction of strain and strain-rate distributions in unknown configurations, and it is difficult to obtain higher accuracy. UB methods ignore elastic strains and stresses. Finite-Difference methods have proved useful in solving thermal problems [1.13] and have some proponents for structural analysis [1.14]. These will be
General introduction to the finite-element method discussed briefly in Chapter 2. Boundary-Element methods have been suggested for small plastic deformation [1.15].
1.3
BASIC FINITE-ELEMENT APPROACH
The use of finite elements for stress analysis of aircraft structures was proposed in 1956, using pin-jointed bars and triangular plates as the elements [1.16], though the fundamental concept was used by ancient geometricians in approximating to a circle by a polygon. The accuracy can, in theory, be made as high as is desired or as time permits, and upper and lower bounds can be established. The circle, for example, must lie between the inscribed and circumscribed polygons (figure 1.1). In modern usage, a limitation is imposed only by the size of the computer memory and the computational time required. As applied to structural mechanics, the fundamental concept of FE formulations is the stiffness K of each element. This relates the force applied to the displacement or deformation produced. Thus, for example, in a simple tensile test a force / produces a stress a in a bar of cross-section A, and an increase oL in the length L. Since the strain e = 8L/L is directly related to the stress by the Young's Modulus E in an elastic body: —
(1.1)
The stiffness can also be deduced from an energy equation using the principle of minimum potential energy or some related variational principle. Because there is only one variable in the simple tension example, this is equivalent to
Fig. 1.1 A circle and the two polygons forming upper and lower bounds.
1.4 General procedure requiring that W, the work done by the external force, and the strain energy U in a specimen of volume Fare related by: dU
dW
d(bL)
d(8L)
=f
(1.2)
For an elastic body in simple tension: ALE 2
AE AE 2L
(1.3)
so: r
fl(8L)
2L
(1.4)
as before. Note that: U =±
(1.5)
Analogous quadratic expressions for strain energy may be derived for all deformation problems. In two- and three-dimensional (2-D and 3-D) elastic problems the Poisson's Ratio v is also involved, as seen later (Section 2.5.1), and in plastic deformation the relationship is not a linear one.
1.4
GENERAL PROCEDURE FOR STRUCTURAL FINITE-ELEMENT ANALYSIS
The structure, which may be a workpiece or other continuum, is first divided into an assemblage of subdivisions or elements, all interconnected at joints or nodes. For example, a beam might be represented by a number of triangles, as in figure 1.2, though in practice many other element configurations are possible [1.2]. See also Section 2.1.7. Fig. 1.2 A beam represented by finite elements.
General introduction to the finite-element method This representation can be considered as a framework, or more conveniently for present purposes and the subsequent discussion of plasticity, as a set of rigid triangular plates. In a further elaboration it could be assumed that each plate was itself capable of deforming uniformly or in some other specified way. The selected assumption defining the field variable throughout the continuum is described as an interpolation function. The equilibrium equations for each of the elements are written down individually, containing the nodal values as the unknown quantities, and the set of simultaneous equations is then solved in matrix form. Once these nodal quantities are known, the interpolation function will provide the complete solution. It is important that an orderly step-by-step procedure should be followed. For a static structure, the sequence of operations is given below. (Note that in the simple one-dimensional examples given in this chapter in which there is only one degree of freedom per node, a symbol with a Greek subscript represents a value at a node; the notation for use in 2-D and 3-D cases will be given later. A symbol with a lower-case subscript in bold type always represents an element quantity.) (i) (ii) (iii)
Discretisation of the structure Selection of the interpolation model Derivation of the stiffness matrix [Ki\ for the equations relating the load or force vectors fi to the displacement vectors d| for each element: f, = [ATjd,
(iv)
(1.6)
Assembly of all the element equations to give equilibrium equations for the whole body: f = [K]d
(1.7)
where d is the global nodal displacement vector and f is the global load vector (v) Modification of the equations as necessary to incorporate boundary conditions, sometimes carried out before the assembly (iv) (vi) Solution of the set of matrix equations. In a linear elastic problem the displacement vector can be found in one step from the applied force vector. Plastic deformation is non-linear and requires incremental or iterative solutions (vii) Computation of the stresses and strains in each element from the nodal displacements, using the interpolation function as necessary This sequence of procedures can be exemplified by reference to a simple tensile test bar.
1.5 Simple application: a tensile test bar
1.5 1.5.1
SIMPLE APPLICATION OF ELASTIC FE ANALYSIS: A TENSILE TEST BAR Discretisation
As the first example, ignore the detailed shape and assume two steps (figure 1.3). There are three elements, of which two require solutions. The displacements dh d2, d3, d4 are unknown.
1.5.2
Interpolation
Assume a linear variation of axial displacement within each element: u(x) = a + bx
(1.8)
where a and b are constants and x is the distance from the left-hand end of the element. Consider element 1, of length Lx\ dx = a + b-0 = a d2 = a + b-Lx\ b = (d2 - dx) ILX
(1.9)
So: (1.10)
= di + (d2 - di) xlLx
1.5.3
Stiffness matrices of the elements
To determine the stiffness of an element, the Hamiltonian principle, commonly known as the principle of minimum energy, is used. For a static system this states that the equilibrium configuration will be one in which the potential energy has a stationary value, usually a minimum. This principle can be applied to the body as a whole, or, as here, to the elements individually. As applied to elastic deformation, the potential energy is the difference
Fig. 1.3 A tensile bar, regarded as three elements.
1
3
2
•
General introduction to the finite-element method tween the total work expended by the relevant external forces and the strain energy stored within the specimen. For pure elastic strain, the latter is recovered in entirety on release of the forces, though in some materials there is appreciable elastic hysteresis loss. Thus for element /, the potential energy: /i = Strain energy of element — Work done by external forces on element = Ul-Wl (1.11) where, for example, for element 1: L
L 1
1
But: €l
= ^1 = *Z*
(L13)
L 1 2
V4VV
V x - X ^7
0
Thus:
This can also be expressed in matrix form using standard matrix algebra with the displacement vectors for element 1:
di = I dJ I ;dl = (dhd2)
(i.i6)
Now:
(1.17) which can be directly multiplied into the vector di to give:
(d, -di, -d, +d2) I 7
I =dl-d2d1-d1d2
+ dl
(1.18)
1.5 Simple application: a tensile test bar
Thus from equation 1.15: 1 -1 -1 1
2L,
(1.19)
£/, = idj [/CJd,
(1.20)
so, by analogy with equation 1.5, the stiffness matrix [K,\ for the /th element is: I -l I
-l
(1.21)
I -l -l I
a
where the stiffness coefficient for element /: (1.22)
a=
Returning now to equation 1.11, the work done by the forces acting on the element / is: W-, = dff,
(1.23)
T
where, for example, f, = (fu, fi 2 ) and/ ia is the force acting upon element / at node a. According to the principle of minimum potential energy, for element /: — =
dda
(f/i-Wj) =0, a =1,2,3,4
(1.24)
dda
so the element stiffness equations are: 0 = [ATJd, - f,
(1.25a)
0 = [K2]d2 -
f2
(1.25b)
0 = [^3]d3 - f,
(1.25c)
Using equation 1.21, these equations may be written in terms of the global displacement vector d = (dh d2, d3, d4)T: 0
0"
gl
0
0
0
0
0
0
0
0
0
0
gl -gl
-gl
f d>\
1
f
\
hi \
d2
/l2
d3
0
\
0/
(1.26a)
10
1.5.4
General introduction to the finite-element method
0
0
-gl
0
/22
-gl
gl
0
/23
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
gl
0
0
0
0
0
gl
0
0
-2
(1.26b)
(1.26c) 2
Assembly of element stiffness matrices
Combining equations 1.26: gl -gl
0 0
-gl
gl+gl -gl
0
Id,
0
0
-gl
0
d2
~g3
d3
gl+g3 ~g3
\
i d* i
g3
1 1
f
/12+/22
\\ (1.27)
/23 +/33
But, for example, / 12 + f22 is the total force acting at node 2, so the right-hand side of equation 1.27 is simply f, the global force vector (fh / 2 , / 3 , / 4 ) T . Hence equation 1.27 may be expressed in terms of the global stiffness matrix [K]: [K]d = f 1.5.5
(1.28)
Boundary conditions
In this example the boundary conditions are very simple and have already been specified in figure 1.3. The element 1 is fixed at the left-hand side and the load / i s applied to the end of element 3. 1.5.6
Numerical solution for the displacements
Let: Al = A3 =200 mm 2 ; Lx = L 3 = 50 mm;
A2 = 100 mm 2 L2 = 100 mm
Et = E2 = E3 = 2x 105 N/mm 2 / = 100 N
1.5 Simple application: a tensile test bar
Then: gl
gl
11
200 x 2 x 10'
50 100 x 2x 105
(1.29)
100 200 x2 x 105 50
gi =
g3 = 8 x 105; g2 = 2 x 105
Hence from equation 1.27: 4 [K]
-4
0
0
4+1
-1
0
0
-1
1+4
0
0
-4
= 2 x 105
-4
4
and so:
f=
h h h
= 2 x 105
n
(1.30)
-4
4
-4
0
0
-4
5
-1
0
(3
-1
5
-4
(3
0
-4
4
-
rf, (1.31) \
The boundary conditions must now be included. Assuming that the bar is anchored at the left-hand end, dx = 0. The force /i is a reaction force and thus does not need to be considered in the equation; the row and column (1) of [K] can thus be eliminated. Furthermore, f2 and/ 3 are zero so: 5
-1
0
d2
-1
5
-A
d3
-4 _0 Writing the equations in full and solving:
4
0 } 0
10
1 °
= 2 x 105
5d2 - d3
+0
=0
-»d
-d2 + 5d3 - 4d4 = 0 0
(1.32)
[ dA\ z
= 5d2
—* da, = 6d2 5
- 4di + 4d4 = 50 xlO" ^d2
= 1.25 x 10^* 4
-> di = 6.25 x 10" = 7.5 x 10^
(1.33)
12
General introduction to the finite-element method
1.5.7
Strains and stresses in the elements dux dx
dT-dl Lx
du2
d^-d?
1.25 x 10"4 = 2.5 x 50 5 x 10"4 IOO
*<> ax
=
jrj? L3
=
1-25 x i o ^ 50
o-j = £ l € l = 2 x 105 x 2.5 x KT6 = 0.5N/mm 2 (j2 = E2e2 = 2 x 105 x 5 x KT6
= l.ON/mm 2
(1.35)
( T 3 = CTX
These are, of course, trivial solutions that can be obtained by inspection, so this lengthy procedure would certainly not be used for so simple a problem. It is however an example that illustrates the way in which problems are solved by the FE method. In the next chapter, we shall consider in greater detail the application of the FE method to small-strain elastic deformation and in particular the structure of various FE matrices, solution methods and other numerical techniques.
References [1.1] Alexander, J.M., Brewer, R.C. and Rowe, G.W. Manufacturing Properties of Materials, Ellis Horwood (1987). [1.2] Zienkiewicz, O.C. The Finite-element Method, McGraw-Hill, 3rd Edn, (1977). [1.3] Irons, B. and Ahmad, S. Techniques of Finite-elements, Ellis Horwood (1980). [1.4] Livesley, R. Finite-elements : An Introduction for Engineers, Cambridge University Press (1984). [1.5] Various, International Conference on Education, Practice and Promotion of Computational Methods in Engineering using Small Computers, Int. Assoc. Computational Mechanics, Macao (1985). [1.6] Nadai, A. Plasticity, a Mechanics of the Plastic State of Matter, McGraw-Hill (1931).
[1.7] Hoffmann, O. and Sachs, G. Introduction to the Theory of Plasticity for Engineers, McGraw-Hill (1953). [1.8] Johnson, W. and Mellor, P.B. Engineering Plasticity, von Nostrand, Reinhold (1973). [1.9] Rowe, G.W. Principles of Industrial Metalworking Processes, Arnold (1977).
References
13
[1.10] Johnson, W. and Kudo, H. The Mechanics of Metal Extrusion, Manchester University Press (1962). [1.11] Avitzur, B. Metalforming Processes and Analysis; McGraw-Hill (1968). [1.12] McDermott, R.P. and Bramley, A.N. Forging analysis - a new approach. Proc. 2nd Nth. American Metalworking Res. Conf., SME, pp. 35-47 (1974). [1.13] Marti, J., Kalsi, G. and Atkins, A.G. A numerical and experimental study of deep elastoplastic indentation. Proc 1st Int. Conf. on Numerical Methods in Industrial Forming Processes, ed. J.F.T. Pittman, R.D. Wood, J.M. Alexander and O.C. Zienkiewicz, Pineridge Press, pp. 279-87 (1982). [1.14] de Arantes e Oliviera, E.R. On the accuracy of finite difference solutions to the biharmonic equation. International Conference on Education, Practice and Promotion of Computational Methods in Engineering using Small Computers, Int. Assoc. Computational Mechanics, Macao IA, pp. 121-9 (1985). [1.15] Brebbia, C. A. Boundary Element Methods in Engineering, Halsted Press (1978). [1.16] Turner, M.J., Clough, R.W., Martin, H.C. andTopp, L.J. Stiffness and deflection analysis of complex structures. /. Aero. Sci. 23,805-24 (1956).
2
Basic formulation for elastic deformation
2.1
TYPES OF ELEMENTS
Discretisation is a fundamental feature of all FE analysis of a continuum [2.1]. In principle many types of element could be used but in practice a small number of types cover almost all metalforming requirements. These are shown in figure 2.1.
Fig. 2.1 A selection of various linear and higher-order elements used in finite-element elastic and plastic analysis.
(d)
(g)
2.1 Types of elements
2.1.1
15
Linear elements
We have implicitly used this very simple type in the tensile bar example (figure 2.2). In general terms, the value of a force applied to a particular node of the isolated element will affect the displacement of all the nodes to an extent determined by the stiffness K:
f=[K]d
(2.1)
or explicitly: 21
K
(2.2) 22
in which, for the sake of simplicity, it is assumed that there is only one element in the body. The stiffness influence coefficient Kap is defined as the force needed at node a to produce a unit displacement at node /3, while all other nodes are restrained. If for example, node 1 is restrained and a unit displacement is assumed at node 2, as in figure 2.3, the necessary force induced at node 2 will be found from: f-Acj-AEe-AE—
d2_AE —-1
(2.
The stiffness influence coefficient K22 is then equal to AEIL. Similarly the stiffness influence coefficient at node 1 due to the force applied at 2 is the reaction force in the opposite direction, so K2X = -AEIL. Fig. 2.2 A simple tensile specimen (a), represented by a linear model (b), showing the forces acting on the central element (c).
f\>di
(c)
Fig. 2.3 The forces and displacements on a single element.
-/-
= 1
16
Basic formulation for elastic deformation
The forces at each node due to a unit displacement at node 1 can also be found, so the full stiffness matrix of the element becomes: +AE/L
-AEIL
-AEIL
+AEIL
(2.4)
AE L
Like many stiffness matrices, this is symmetric. It has already been deduced in a different way in Chapter 1, equation 1.21, where it was used to solve the first tensile specimen problem.
2.1.2
Plane-strain triangular elements
Triangular elements (figure 2.1a) are widely used for plane strain and axisymmetric problems. Assuming the strain to be uniform in an element:
U\
\ du2
e22
(2.5)
dx2
\
dUi
du2
dx2
dXi
I
J
where U\ and u2 are the components of the displacements in the X\ and x2 directions, en, e22 are direct strains and y12 is an engineering shear strain.
2.1.3
Linear quadrilateral elements
Very often the geometric shape can conveniently be filled with rectangular elements (figure 2.1b). These are less stiff than the triangles in plane strain; in an extreme example, under constant-volume conditions the apex of a triangle can move only parallel to its base when the base nodes are fixed.
2.1.4
Higher-order (quadratic) elements
Because of this problem of stiffness it is often desirable to use higher-order elements (figure 2.1c,d). Greater flexibility is obtained, at the expense of larger matrices and longer solution times, by introducing mid-side nodes. The edge of an element is then, in general, a parabola instead of a straight line. Quadrilateral elements can also be provided with mid-side nodes, producing eight-node elements. Again flexibility is improved at the cost of memory size and computing time. The advantage is usually less for quadrilaterals than for triangles. In some circumstances even the use of two intermediate nodes on each side
2.1 Types of elements
17
may be desirable to model complex distortion accurately. Such third-order or cubic elements are seldom required.
2.1.5
Three-dimensional elements
The most popular element types for 3-D plastic deformation are rectangular bricks (figure 2.1e,f). These may have eight nodes, one at each corner, and can be more flexible with additional mid-side nodes. The size of matrix is a critical feature of 3-D analysis. Even a coarse mesh of 10 elements on each side requires 1000 elements and the number of nodes must also be controlled. Tetrahedral elements are also widely used (fig. 2.lg).These can often be fitted around corners. Different types of element may be incorporated in a single network, for example to model complex shapes more easily.
2.1.6
Size of elements
Generally speaking, the accuracy of FE solutions improves as the number of elements is increased. The number is normally limited by the size of the computer memory and the time for solution of the simultaneous equations. As in the example of figure 1.2, the general pattern of discretisation is quickly established with relatively few elements. Beyond a certain limit, the computing cost of refining the mesh rises rapidly without corresponding gain in useful information. For plane-strain or axi-symmetric plasticity problems, it is seldom desirable to use more than 1000 elements. In 3-D though, this gives a rather coarse mesh and even a ten-fold increase in the number of elements only halves the linear mesh dimensions, so definition of detail presents serious problems in 3-D. To save computer space and time, it is usual to combine large-element meshes in regions of secondary interest with finer meshes in critical zones. For example figure 2.4 shows a mesh prepared for analysis of indentation. The region around the corner of the indenting punch must be well defined as it is known that the deformation is symmetrical about the centre line and occurs mainly in a narrow band with steep lateral strain gradients, especially near this corner. Fig. 2.4 Fine and coarse-mesh elements used for indentation.
Basic formulation for elastic deformation
18
Small mesh size also usually improves the convergence of an iterative solution.
2.1.7
Shape and configuration of elements: aspect ratio
It is convenient to use simple triangular and quadrilateral elements, usually chosen with an aspect ratio of about unity. This provides satisfactorily uniform deformation and allows simple interpolation functions to be used. Nevertheless, if the pattern of deformation permits, it is possible to economise on computer storage and time by using long, thin elements instead of small squares in certain regions. This can be seen in figure 2.4 and another example is given in figure 2.5 for open-die extrusion. In many structural analyses, the most efficient mesh may be much more complex, as shown in figure 2.6 [2.1]. Such meshes can be generated by one of the commercial optimising algorithms. This is very useful but is not strictly necessary for many metalforming analyses.
2.2
CONTINUITY AND EQUILIBRIUM
The basic FE equations for plastic or elastic deformation are simple, relying on equilibrium of forces at every node and the compatibility of strains between elements. In a linear problem, for example in elastic deformation of metals, the displacement vector can be found directly from the force vector, or the reverse, using the stiffness relationship for each element given in equation 2.1. Fig. 2.5 A non-uniform mesh used for extrusion analysis.
0 OOOOOC
o
OOOOC
O O OO O O O O O C
2.3 Interpolation functions
19
As explained in Chapter 1, Section 1.5, a stiffness relationship can be established for plastic deformation using the principle of minimum potential energy. Because the FE solution is obtained only at the nodes, there will be discontinuous changes from one node to the next. It is therefore necessary to introduce interpolation functions to describe the behaviour of the solution across an element while still satisfying the equilibrium conditions [2.2].
2.3
INTERPOLATION FUNCTIONS
2.3.1
Polynomials
These functions should be as simple as possible, whilst being compatible with the required accuracy and detail of the solution. It is usual to select a polynomial function, which can often be simply a linear relationship: N(x) = a + bx + ex2 + • • • + px r
(2.6)
The common linear approximation for 2-D is: N(xhx2) = a + bxxi + b2x2 (2.7) It is usual to apply the same interpolation function to both the geometric description of the element and the displacements. The elements are then known as isoparametric, but sometimes one interpolation function may be of higher order than the other. We shall not consider these here.
2.3.2
Convergence
The interpolation function must satisfy the requirements of convergence or the finite-element solution will not converge on the true solution as the element size is reduced. Fig. 2.6 A mesh for analysing a block containing a large hole.
20
Basic formulation for elastic deformation
The first requirement is that the interpolation function must be continuous. Polynomial functions satisfy this condition. Secondly, uniform states of the field variable and its derivatives must be represented when the element size reduces to zero. In particular, zero strain and constant strain must be compatible with the assumed displacement model. Thirdly the variable and its derivatives (up to one order less than the highest found in the functional I(N)) must be continuous at element boundaries and interfaces. In solid mechanics this means that there can be no discontinuities or overlaps between elements, and the slopes of beams, etc, must be continuous. The elements and functions normally used in plasticity satisfy all these conditions. In other words they are 'complete'.
2.4
DISPLACEMENT VECTOR u AND MATRIX [B]
The first requirement for a solution, once the element mesh has been established and the interpolation function has been selected, is to define the displacements of the nodes and from these to deduce the strains in each element. The basic technique can be explained with reference to simple triangular elements. The displacement vector u of any point (xh x2) within the element has components uh u2 parallel to the two axes. Assume these to be linear functions of the co-ordinates: ut = at + buXi + bi2x2 , / = 1,2
(2.8)
This displacement must be equal to the nodal displacement when evaluated at the nodal positions. The equations thus obtained are presented and solved in Appendix 1. A typical expression of one of the coefficients is given in equation A1.10: (*3i-*2i)dn + {xn-x3l)d2l + (x21-xn)d31 I
(2.9)
where A is the area of the triangle 123 in figure 2.7, xu is the /th co-ordinate of node / and du is the displacement of node / in the xt direction. Substituting for the coefficients in equation 2.8: Wi = —
I *21*32-*22*31 + (*22-*32)*l + (*31-*2l)*2 I ^1/
2A \
w
+ +—
LA
I
x
xx\
I XUX2l ~X12X2l + (*12-*22)*1 + (*21-*ll)*2 I d3i
\
I
2.4 Displacement vector u and matrix [B]
21
or: (2.10 continued)
where Nj is the interpolation function of position relating to node /. As shown in Appendix 1, the strain vector € is given by:
6 =
*32-*12
0 xn-x3i
(d"\
0 xn-x3l
•^12-^22
•*32-*12
X2i~Xn
0
dn d2i
X12-X22
d12 d31
or 6 = [B]d
(2.11)
The matrix [B] is generally a function of position, though for the constant-strain triangle of this example it is independent of position within the element. The [B] matrix is of great importance in all finite-element analysis.
Fig. 2.7 Deformation of a simple triangular element. X
2
i
22
Basic formulation for elastic deformation
2.5
THE [D] MATRIX OF ELASTIC CONSTANTS
2.5.1
General expression
We consider first the [D] matrix in terms of linear elastic constants. This will be developed for non-linear plastic deformation in Chapter 3. In one dimension (1-D) Young's Modulus E is the ratio of stress <xto strain e: e'
6n
(2.12)
E
In 2-D the Poisson's Ratio p must be included: (2.13) and similarly in 3-D: On
O- 22
CT33
(2.14) v— E E E The shear strain in 2-D can be derived by considering a cube subjected to a shear stress (Appendix 7). For example: en =
v
2(1 + v)
o-12
(2.15)
where G is the elastic Modulus of Rigidity. Collecting these expressions into a matrix equation, the isotropic elastic modulus is given by the equations:
' eA
1
e22
—V
1
—p
—v 1 —V
0
0
0
—p
0
0
0
1
0
0
0
O-33
0
0
O" 12
0
O-23
0
0
0
2(1 + p)
723
0
0
0
0
2(H
713
0
0
0
0
0
712
~E
1 au\
—v
(2.16)
2{\ + v)
In practice, this relationship is required in the form: o- = [D]e
(2.17)
2.5 The [D] matrix of elastic constants
23
In Appendix 2 it is shown that: crn\ O-22
or 3 3
cr 12
1+v
O-23
(2.18)
i-v
V
V
\-2v
\-2v
\-2v
V
1-v
V
\-2v
\-2v
\-2v
V
V
1-v
l-2i/
1-2 v
1-2 v
0
0
0
—
0
0
0
0
0
—
622
712
0
723
2
0
0
0
o
0
1
.713
There are three important special conditions often met in 2-D analysis that permit the [D] matrix to be simplified. These are the conditions of plane stress, plane strain and axial symmetry.
2.5.2
Plane elastic stress
Under plane-stress conditions: (X33 — (T13 — (T23
=
(2.19)
0
Furthermore, since the two shear stresses above are zero: (2.20)
713 = 723 = 0
and equation 2.18 can be re-written immediately:
/ \ / on \
w O- 2 2
E 1+v
1-v
V
1/
\-2v
l-2i/
l-2i/
V
1-v
1/
1-2 v
l-2i/
1-21/
0
0
0
But: 0-33 =
(1 +1/) (1-21/)
[ven + ve22
n
le611\
0
^22
\j
1
y.
(2.21)
633
W (2.22)
Basic formulation for elastic deformation
24
so: (2.23)
1-v Substituting for e33 into equation 2.21: \ 2
v2
1-v
v
1-21/
(1-v) (l-2i/)
1-2^/
(1-v) (l-2i/)
(l-i/) (1-2*/)
l-2i/
(I-*/) (1-2J/)
a22
611
0
^22
1 2".
0-12
1-1/
0
2.5.3
0
VYuj {
i
1-v 1 1-v
^22
0
7l2
(2.24)
Plane elastic strain
Under plane-strain conditions: ^33 = 7l3
=
(2.25)
723 = 0
Furthermore, since the two shear strains above are zero: O"13 =
<723 =
(2.26)
0
and since the strain €33 is zero, the corresponding normal stress does not enter into the strain energy expression and equation 2.18 can be re-written as: \
On
1-v
V
1-2 v
l-2i/
v
1-v
1-2 v
1-2 v 0
\
2.5.4
0 622
—
(2.27)
712
Axial symmetry
If xi is identified with the radial direction, x2 with the circumferential direction and x3 with the axial direction, the conditions of axial symmetry require that: 712 = 723 = 0
(2.28)
2.6
25
The element stiffness matrix
Hence: (2.29) and equation 2.18 becomes: /
1-v
\ a22
E
v
1-v
l-2i/
1-21/
1-21/
l + i/
l-i/
0-33
\
2.6
&13 I
1-2 v
1-21/
0
0
0
€2 2
(2.30) 0
0
—
w
FORMULATION OF THE ELEMENT STIFFNESS MATRIX
As in equation 1.11, the potential energy / is found from the difference between the work done by external forces and the strain energy stored. For example, in plane stress (<x33 = o-o = o-23 = 0): I = U-W =
^r
jf (^n !! + ^22^22
(2.31)
6
- f22d22
~
in which//, denotes the component of force in the Jt; direction acting at node /. In matrix form the expression for element i is:
(2.32)
[£>,]«! dV.-dif,
= ydiM[BI]T[A][BJdViJdI-dTfl Differentiating with respect to each of the components of displacement and equating the derivatives to zero leads to a set of conditions for the minimum of the potential energy:
J
i-f,
(2.33)
or: (2.34)
26
Basic formulation for elastic deformation Thus the elastic stiffness relationship for element / may be written as: [*i]d, = fi
(2.35)
[*,]=/[B I ] T [D l ][B l ]dV l
(2.36)
where:
The derivation of this stiffness matrix for the case of a constant-strain triangle deforming under plane-stress conditions is given in detail in Appendix 4. For a constant-strain triangle, the integrand in equation 2.36 is the same throughout the element and so: [*,] = s-At [B,]T[A] [Si]
(2.37)
where s is the assumed thickness of all the elements.
2.7
FORMULATION OF THE GLOBAL STIFFNESS MATRIX [K]
2.7.1
Assembly of element matrices
The vector fj in equation 2.35 contains all the components of nodal force acting upon element / and, for an isolated element, is simply the vector of applied nodal loads. However, in general, the element will not be isolated, and in this case the forces acting upon element / will be the sum of the applied loads and the reactions from any adjoining elements. We do not know, and often do not need to know, the values of these reactions, but this does not matter because it is found that they cancel (action and reaction being equal and opposite) when the value of force acting at a given node is summed over all the elements containing that node. In this process, the individual element stiffness matrices are assembled into a global stiffness matrix [K], the coefficients of which determine the influence of a component of displacement of one node in the mesh upon a component of force applied at another node.These coefficients are simply the sums of the corresponding coefficients in all the element stiffness matrices. An example of the assembly process was given in Chapter 1 (Section 1.5.4). The global stiffness equations may be written as: [K]d = f
(2.38)
where the global displacement vector d is the vector of the components of displacement of all the nodes in the mesh, and the global force or load vector f contains the components of force acting at all the nodes. In an elastic deformation problem [X] is usually linear (though large strains in rubbers for example are recoverable and therefore elastic, but are not linearly related to stresses). There is considerable interest in the solution of largedeformation elastic problems, but these do not concern us here.
2.7 The global stiffness matrix (K]
2.7.2
27
Properties of [K]
The global stiffness matrix [K] has several properties that have important consequences for the computer implementation of the FEM. Although we are at present assuming that the deformation is purely elastic, these properties are also exhibited by the stiffness matrix for small-deformation plasticity derived in the next chapter, and by the finite-deformation elastic-plastic stiffness matrix to be introduced in Chapter 5. Symmetry
Since the individual matrices [Z)J are symmetric, it can readily be seen, by transposing equation 2.36, that the element stiffness matrices [Ki\ also have this property. Consequently [K] is also symmetric because the element stiffness matrices are assembled into the global matrix in a symmetric manner. Since [K] is symmetric, only about half of the coefficients need be calculated and stored. As will be seen in the next chapter, the constitutive matrices used in elastic-plastic analyses are also symmetric and so lead to symmetric stiffness matrices. Positive semi-definiteness
[K] is positive semi-definite since, for all non-zero vectors v, the quadratic form vT[/T|v (a scalar quantity) is greater than or equal to zero. This property can be deduced from the principle of minimum potential energy that forms the basis for the derivation of the element stiffness matrices [#J. It can easily be shown that any semi-definite matrix is singular, and so does not have an inverse. The stiffness matrix in equation 2.4 is an example of this. The singularity of [K] is a consequence of one of the requirements for the completeness of the FE model. In simple terms, the displacement vector obtained by adding any rigid-body translation or rotation to a solution of equation 2.38 will also be a solution of this equation. Since the basic stiffness equations permit an infinite number of solutions for the displacement vector, it is necessary to specify values for a minimum number of independent components of nodal displacement (five for 3-D problems). In metalforming analysis, these will be determined in the normal course of things by the nature of the process model, such as the existence of dies or planes of symmetry. The specification of values for certain components of displacement is equivalent to converting the set of stiffness equations into a lower-order set for which the matrix of coefficients is positive definite (that is, has quadratic forms that are strictly greater than zero.) It should be remembered however, that any computer representation of the stiffness matrix will always be an approximation. An FE program may therefore succeed in obtaining a solution displacement vector even when insufficient num-
Basic formulation for elastic deformation
28
bers of displacement components are prescribed. The value of a solution under such circumstances is questionable. Bandedness
Since the global stiffness matrix is assembled on an element by element basis, the value of force at a particular node can depend only upon the displacement of nodes that belong to the same element as the one under consideration. Thus it is found that each row of [K] contains many zero coefficients. If the nodal entries in the global vectors d and f are ordered correctly, it is possible to confine the non-zero coefficients of [K] to a band either side of the leading diagonal. The zero coefficients outside the band do not have to be stored, and they do not have to take part in any subsequent calculation. The extent to which the width of the band can be reduced is obviously crucial Fig. 2.8 Non-zero coefficients in the global stiffness matrices for typical 3-D meshes with the diagonal bands bounded by the dashed lines: (a) 4x4x4 nodes, (b) 6x6x6 nodes, (c) 4x6x26 nodes, (d) square at top left-hand corner of (c) enlarged to show sparse nature of band. Each square symbol in these figures represents the 3x3 sub-matrix corresponding to the three degrees of freedom at a node. (a)
N
nodes
2.7 The global stiffness matrix (K]
29
in determining the maximum size of problem that can be solved by an FE program, and in reducing the time of computation. Figure 2.8 shows the patterns of non-zero coefficients in the global stiffness matrices for some typical 3-D meshes. In these examples, the order of the nodes in the global vectors is not optimal. These diagrams give an indication of the reductions in computer storage and computational time that are possible as a result of a suitable re-ordering of the nodal vectors. Sparseness of diagonal band
Even when [K] has been arranged to have the narrowest possible width of band, it is often found that the band is sparse, that is, contains many zero coefficients. Whereas the band-width is influenced by topology of the mesh (the types of element and their arrangement) and the ordering of the nodes, the number of non-zero entries in a row of [K] is independent of the nodal ordering. The sparse nature of the diagonal band can be seen clearly in figure 2.8d. Not all techniques for solving the stiffness equations are able to exploit this sparseness.
30
Basic formulation for elastic deformation nodes
2.8 General solution methods
2.8
31
GENERAL SOLUTION METHODS FOR THE MATRIX EQUATIONS
As mentioned in the previous section, the solution of equation 2.38 must incorporate a certain number of prescribed components of displacement. We will consider first of all general solution methods for non-singular sets of matrix equations in which all the components of displacement are unknown, and examine ways of incorporating values of displacement into these solutions later. Matrix equations such as 2.38 may be solved by direct or by indirect means. Direct methods obtain a solution by calculating the inverse relationship to equation 2.38; indirect methods iteratively modify a starting guess for d until a vector is obtained that satisfies the matrix equations.
2.8.1
Direct solution methods Gauss-Jordan elimination
This is the method usually adopted when solving a small set of simultaneous equations by hand. By a sequence of row operations - in which multiples of one equation are added, coefficient by coefficient (including the right-hand side), to another - the matrix equations 2.38 are converted into an equivalent set that has the same solution vector: [T]d = r
(2.39)
The new matrix [7] has zeros everywhere except on its leading diagonal. The components of the displacement vector may then be obtained immediately by dividing each of the components of the right-hand side vector r by the corresponding diagonal term of [7], Gauss-Jordan elimination is not often used in FE work, partly because the number of computational operations involved is about 50% more than those required for a partial elimination scheme [2.3], but, more importantly, because the whole elimination has to be repeated if a solution is required for a new force vector. This makes the method unsuitable when multiple load cases and iterative techniques are considered. Gaussian elimination and back-substitution As in the previous method, row operations are performed on the stiffness matrix and force vector in order to obtain a set of matrix equations that have the same solution vector as the original ones: [U]d = s
(2.40)
This time, [U] is an upper-triangular matrix, that is, one having non-zero coefficients on and above the leading diagonal and zeros below the diagonal. The back-substitution part of this method uses equations 2.40 to calculate the components of d in reverse order:
32
Basic formulation for elastic deformation
da= - M
sa-
Z
Ua0dp) ,a = n,
(2.41)
in which n is the number of degrees of freedom in the FE discretisation and the summation is taken to be zero when a = n. It can be seen that when the time comes to evaluate da, all the values of the displacement components from da+\ to dn contained in the summation are already known. The method of Gaussian elimination and back-substitution is described in detail in Appendix 5. It requires fewer computational operations than a complete Gauss-Jordan elimination, but for general non-symmetric matrix equations a complete re-calculation is still required to solve for each new right-hand side vector. It shall be seen that this is not the case for the symmetric matrices encountered in FE metalforming analysis. Choleski LU decomposition Any square, non-singular matrix can be expressed as a product of a lowertriangular matrix [L] and an upper-triangular matrix [£/]. Thus we can write equations 2.38 as: [L] [U]d = f
(2.42)
There are infinitely many LU decompositions of a given matrix. However, if n entries are specified, the decomposition becomes unique. In most Choleski decomposition schemes, the n diagonal coefficients of [L] are all chosen to be unity. Making the substitution: s = [U]&
(2.43)
[L]s - f
(2.44)
we obtain: Since [L] is in lower-triangular form with unit diagonal coefficients, equations 2.44 may be solved by forward-substitution:
sa=fa-
0=1
and then the values of the components of s may be used to solve equations 2.43 by back-substitution as in the previous method. The advantage of this method is that, once the stiffness matrix has been decomposed in this way, solutions can be obtained for new force vectors with little extra computation (of the order of n2/2 multiplications and additions for the forward substitution, compared with aproximately n3/3 multiplications and
2.8
General solution methods
33
additions for the elimination phase of the previous method, back-substitution being common to both). The LU decomposition may be performed using the Crout algorithm [2.3]. However, since the stiffness matrix is symmetric, it can be shown that [L] is simply the transpose of [£/], but with the entries in each column divided by the diagonal coefficient:
L«, = -?£=-, 0*««*
(2-46)
In addition, the upper-triangular matrix [U] in the LU decomposition with unit diagonal [L] is precisely the same as the upper-triangular matrix obtained during the Gaussian elimination procedure (providing the rows of the latter matrix are not already normalised by dividing through by the diagonal coefficients). Thus for the symmetric stiffness matrices encountered in metalforming analysis, Gaussian elimination is equivalent to a Crout decomposition, although there may be slightly less round-off error accumulated in the latter process [2.3]. Whichever technique is used, only the coefficients of [U] need to be explicitly calculated and stored, since those of [L] can easily be obtained from these as required, whether for the first solution, or any subsequent re-solution. It is a simple matter to modify the above solution methods so that only those coefficients lying within the non-zero diagonal band of [K] are involved in the computation. Unfortunately, during both the Gaussian elimination and Crout decomposition procedures, the zero coefficients within the band tend to be filled up with non-zero ones, so these methods cannot take advantage of the sparseness of the band itself. Direct methods can however be modified so that only a small part of the band of the stiffness matrix need be in computer memory at any one time, the rest of the matrix being saved on secondary disc storage (Section 6.6.3).This permits a much larger stiffness matrix to be accommodated than would be the case if all the non-zero coefficients had to be in computer memory at the same time, though at the expense of increased program running time.
2.8.2
Indirect solution methods
Whereas the direct methods discussed in the previous section are almost certain to produce a solution of some sort after a predetermined amount of computation, it is not possible to determine beforehand how many iterations will be needed in order to converge to a solution using indirect techniques. For some types of problem indirect methods can be very efficient indeed; for others the computational effort may be many orders of magnitude greater than that expended during a direct solution. Moreover, since the stiffness matrix is repeatedly used in a large number of iterative steps, the storage of the stiffness matrix on disc would entail a prohibitively-large number of time-consuming disc accesses. It is
34
Basic formulation for elastic deformation
therefore essential that all the non-zero coefficients of this matrix be stored in the main memory of the computer. This effectively limits the size of problem that can be examined. For these reasons, indirect solution methods will not be considered in great detail here. However, since the effectiveness of these methods depends very much on the nature of the matrix equations being solved, and since larger and cheaper computer memory chips are continually being introduced, the reader is encouraged to experiment with the use of these techniques for particular problems. Also, it should be noted that iterative solutions are ideal candidates for parallel processing techniques, so the arrival of parallel processing hardware, such as Transputer boards, to be used in conjunction with computer workstations may cause a complete re-appraisal of iterative methods of solution. The problem of finding the solution of equations 2.38 is equivalent to the problem of finding the minimum of the function h of n variables where: ft(d) = —d
T
[#]d-d T f
(2.47)
Since [K] is positive semi-definite, this function does have a minimum, and since [K] is symmetric, the condition for a minimum of the function: |L[tf]d-f = 0
(2.48)
is precisely the condition that the vector d at which the minimum occurs is also the solution of equations 2.38. Multi-dimensional minimisation techniques search for the minimum of a function by a succession of one-dimensional minimisations along lines in n-dimensional space. The group of techniques known as conjugate-gradient methods, choose the directions for the line minimisations in order to optimise the searching procedure [2.3]. Even so, a large number of steps may be required unless [K] is approximately proportional to the identity matrix. This fact is exploited by a hybrid technique particularly suited to the solution of sparse matrix equations, the incomplete Choleski conjugate-gradient method [2.4]. In this procedure, an approximate Choleski decomposition of [K] is performed during which a coefficient of the upper-triangular matrix [U*] is only calculated if the corresponding coefficient of [K] is non-zero, otherwise it is assumed to be zero. This considerably reduces the amount of computation and storage involved in the decomposition. The matrix [L*][£/*], where [L*] is calculated analogously to equation 2.46, is only approximately equal to [K] and so cannot be used directly to solve for the vector d. However, the equations: [K*]&* = f*
(2.49)
where: [K*] = [L+Y^KRU*]-1
(2.50)
2.9 Boundary conditions
35
and: f* = [L*]-1 f
(2.51)
d* = [t/*]d
(2.52)
have the solution: Since [K*] is an approximation to the identity matrix, conjugate-gradient methods quickly converge to the solution of equation 2.49. [U*] and [L*] are triangular matrices, so the inverses in equations 2.50 and 2.51 may be computed with little effort, and the required solution vector d may easily be obtained by inverting equation 2.52.
2.8.3
Iteration to improve a direct solution
Even if iteration is not used to produce the solution itself, it can be used to good effect to counter the accumulation of round-off error in a direct solution. Suppose equations 2.38 have been solved to produce an estimate of the solution vector which, due to round-off error, differs by an unknown vector 8d from the correct solution. Substituting the approximate solution d 4- 8d into equations 2.38 gives an approximate value f+ for the force vector: [K](d + 8d) = f+
(2.53)
Substituting from equation 2.38: f + [K]8& = f+
(2.54)
and rearranging: [K]8d
=f+-f
(2.55)
Since the right-hand side vector of equations 2.55 is known, these equations can be solved to obtain the correction that needs to be applied to the original estimate of the displacement vector. This solution will require very little extra computation, providing [£/], the upper-triangular decomposition of [K\, has been saved. Naturally, due to round-off effects, the value of 8d thus obtained will itself be slightly in error, and the process of iterative correction may be repeated. However, it is quite likely that a single correction will give sufficient improvement in the accuracy of the displacement vector, particularly if the vector on the right-hand side of equations 2.55 is calculated using double-precision arithmetic.
2.9
BOUNDARY CONDITIONS
In the FE analysis of structural problems, it is usually required to find a set of nodal deflections that correspond to a specified set of nodal or distributed loads given that the displacement of supported nodes is zero in particular directions.
36
Basic formulation for elastic deformation
However, in metalforming analysis, the deforming loads are rarely known beforehand, whereas the components of displacement of certain nodes will be determined by the boundary conditions of the problem under consideration. For example, if a node is in contact with a die during a forging operation, the displacement of that node perpendicular to the die surface will be the same as that of the die itself. (The displacement parallel to the die surface may not be known, depending as it does upon a number of factors, including the frictional conditions of the interface between the die and the workpiece.) Although the applied nodal forces are not generally known, the resultant force will be known at many nodes, specifically those inside the body and those on the free surfaces of the workpiece. At these nodes, the resultant force is zero and it is the displacement that must be determined. At the start of the analysis then, most of the components of f are known to be zero, but some components are unknown.The components of d corresponding to the latter will be specified by the boundary conditions of the problem. The solution of the stiffness equations must therefore take these known components of displacement into account and calculate the corresponding reactions, as well as all the other components of displacement. There are several ways of accomplishing this. One method, which involves multiplying certain coefficients of [K] by very large numbers is described in reference [2.5] and is incorporated into the BASIC program listed in Appendix 11. Appendix 6 describes another method, one which modifies the Gaussian elimination and back-substitution procedure. This appendix also considers the more complicated situation in which the specified components of displacement are not parallel to any of the global axes.
2.10
VARIATIONAL METHODS
Before going on to consider small-deformation elastic-plastic FE analysis, it is worthwhile mentioning some of the other numerical techniques that can be used to study plastic deformation problems.
2.10.1 Variational method of solution in continuum theory Most plasticity analysis has used a variational approach, developed from conventional continuum methods [2.6]. In these, differential equations are set up, governing the behaviour of infinitesimal domains. An integral /is thus defined, which can sometimes be determined directly. If not, the variational principle is invoked, stating that the correct solution is the one that minimises the value of /. Since several independent variables may be involved, / is expressed as a functional, that is, as a function of several other functions, for example:
2.10 Variational methods
37
x2
(2.56) x1 where 4>' and " are the first and second derivatives of the function (f)(x). The problem is then to find the function 4>{x) which gives a minimum (stationary) value of /. Usually the function / has a clear physical meaning, for example it may be the potential energy, which is a function of both displacement and position. In some instances, it is possible to determine both maximum and minimum values of two functional /, and thus to produce upper and lower bounds to the solution of a more complex problem, but this is unlikely in plasticity analysis.
2.10.2 Approximate solutions by the Rayleigh-Ritz method The problem can be stated formally as the requirement to minimise the functional I(4>) in a volume V, subject to set boundary conditions on the boundary area S: I = ]>(*, >, 4>')dV
(2.57)
According to the Rayleigh-Ritz method [2.7], an approximate solution is assumed for the field variable: n 1=1
where g;(x) are known functions defined over the volume and surface. The parameters ct have to be determined, to satisfy the condition: = 0,i=l,n (2-59) dct which leads to n simultaneous equations. The accuracy of the solution depends on the choice of the 'trial' functions gt. These must be part of a set that produces a convergence to the correct solution, and they must be continuous up to the necessary degree of differentiation in the functional, etc. Usually these functions are in the form of a polynominal or a trigonometric expression. The Rayleigh-Ritz method resembles the general finite-element approach.
2.10.3 Weighted residuals The method of weighted residuals [2.8], used to obtain approximate solutions to linear and non-linear differential equations in continuum problems, does not require a knowledge of the functional. The statement of the problem takes the form: F(4>) = G((j)) in a volume V
(2.60)
38
Basic formulation for elastic deformation
subject to a set of boundary conditions on the surface S. The field variable itself is again approximated, as in equation 2.58. The next step is to define a residual or error functional R, such that: R(cj)) = G(0) - F(>)
(2.61)
and R has to be minimised for the best solution. A weighted function of this residual, wp(R) is assumed to satisfy the acceptable residual condition. If the function is chosen so that p(R) = 0 when R = 0, then the corresponding function cf)(x) will be equal to the exact function 4>(x). The trial function 4>(x) is chosen to satisfy the boundary conditions and the acceptably small residual is given by the integration over the whole volume: lwp(R)dV=0 (2.62) There are several ways of using this approach, of which the Galerkin method is most widely used [2.9]. Another uses the 'least squares', by weighting the squares of residuals.
2.10.4 Galerkin's method In this method the weighted functions are chosen to be the known functions gt and the n separate integrals are each equated to zero: I gt(R)dV=
0, i = l,n
(2.63)
There are thus n simultaneous equations to be solved for the n unknown coefficients C/.
2.11
THE FINITE-DIFFERENCE METHOD
Mention should also be made of a completely different method of solving nonsteady-state problems. The basis of this is to approximate a first derivative by a small finite step. For example if the quantity T, which might be temperature, varies with time:
f^^° dt
(2.64) At
where: T1 = T{t + ViAi) ; T° = T(t - ViAt) (2.65) and A Ms a small time interval. This is known as the finite-difference method [2.10]. A known initial condition is used to find the solution at time (f + At), and the solution proceeds in small sequential steps. This method will not be used for isothermal or adiabatic plasticity problems but it is useful for heat flow analysis and can be used in the temperature part of a coupled thermo-mechanical solution.
References
39
References [2.1] Zienkiewicz, O.C. The Finite-element Method, 3rd Ed., McGraw-Hill (1977). [2.2] Rao, S.S. The Finite Element Method in Engineering, Pergamon (1982). [2.3] Press, W.H., Flannery, B.P.,Teukolsky, S.A. and Vetterling,W.T. Numerical Recipes, Cambridge University Press (1986). [2.4] Kershaw, D.S. The Incomplete Choleski-Conjugate Gradient method for the iterative solution of systems of linear equations. /. Comp. Phys. 26, 43-65 (1978). [2.5] Cheung, Y.K. and Yeo, M.F. A Practical Introduction to Finite Element Analysis, Pitman (1979). [2.6] Strutt, J.W. (Lord Rayleigh) On the theory of resonance. Trans. Roy. Soc. A161, 77-118 (1870). [2.7] Ritz, W. Uber eine neue Methode zur Losung gewissen Variations. /. ReineAngew. Math. 135, 1-61 (1909). [2.8] Mikhlin, S.C. Variational Methods in Mathematical Physics, Macmillan (1964). [2.9] Galerkin, B.G. Series solution of some problems of elastic equilibrium of rods and plates. Vestn. Inzh. Tech. 29, 897-908 (in Russian) (1915). [2.10] Southwell, R.V. Relaxation Methods in Theoretical Physics, Clarendon Press (1946).
3
Small-deformation elastic-plastic analysis
3.1
INTRODUCTION
The methods described in Chapters 1 and 2 are generally applicable though the discussion so far has been related largely to linear elastic problems. In these problems the [D] matrix does not change during deformation and the changes in the [B] matrix due to the changes in nodal co-ordinates are small. In plastic deformation, and especially in metalforming, the strains may be very large. A change of 30-40% in diameter of a drawn wire or height of a forging, for example, can often be required. In extrusion the strains may be much greater still, reaching logarithmic strains of 5 or more. This introduces very severe local distortion and we shall consider the problems involved later, in Chapters 5 and 6. Even when the deformation involves only a small amount of plastic strain, the yield stress will change in a non-linear manner. The metal deforms elastically at first with essentially constant values of Young's Modulus and Poisson's Ratio, but as soon as the stress reaches the yield value, plastic deformation occurs with a much lower effective modulus. The yield stress itself increases with strain due to work hardening, and stress is no longer proportional to strain, but is related to the strain increment and, at least for hot working, also to the strain rate. In this chapter, we shall consider only isothermal plastic deformation at temperatures such as the normal ambient, well below the range of hot working. To avoid serious mesh distortion problems we shall also assume that the deformation is moderate, say less than 10%.
3.2 3.2.1
ELEMENTS OF PLASTICITY THEORY Yielding
Figure 3.1 shows two representative stress/strain curves, respectively for commercially pure aluminium, which is widely used as a model material, and for an aircraft alloy 7075.
3.2 Elements of plasticity theory
41
These results are obtained by compression testing, since tensile tests are limited in total extension. For FE analysis they can be regarded as flow stress curves, showing how the flow stress Yincreases with plastic strain. They are commonly represented by simple constitutive equations, for example: (3.1)
Y=
where ep is the accumulated plastic strain to be defined later. More accurate analysis requires better fitting of the curves, using polynomials, or more complex expressions. For example, the yield stress of commercially-pure aluminium may be expressed as a logarithmic and exponential function of plastic strain for small strains, and as a linear function for larger values: Y= 162A + 2Hn(6p+0.00465) +
, ep<0.7
Y= 164.8 +72(e p -0.7),
ep>0.7
(3.2)
As well as knowing the numerical value of the flow stress under these highly controlled and simplified conditions, it is essential to know how the material will react under a combination of stresses. For this purpose the well-known von Mises yield criterion is used. In terms of principal stresses (defined as the direct stresses on planes where the shearing stress is zero), this criterion is usually written: (o-!-c72)2 + (cr2-o-3)2 + (o-3-o-!)2 = 2Y2 = 6k2
(3.3)
in which k is the yield stress in pure shear. The von Mises criterion is often interpreted as being determined by the shear strain energy. Fig. 3.1 Representative stress/strain curves for (a) commercially-pure aluminium and (b) aluminium alloy 7075.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
42
Small-deformation elastic-plastic analysis
3.2.2
Deviatoric and generalised stresses
In plastic deformation, which is essentially a process of shear, we are concerned less with principal stress and more with the shear stresses, or in general terms with the deviatoric stresses a\j. These are defined in contradistinction to the hydrostatic stress or mean stress o*1, which, being uniform in all directions, causes only an elastic volume change, with no change in shape: (T —
((Tii + °22 + ^33)
(3.4) 1 where the suffix notation implies summation of all the terms. The deviatoric stress is then the difference between the total stress and the hydrostatic stress:
(3.5)
The Kronecker delta 5,, is defined to be equal to one if i=j, and equal to zero if i^j. Thus a deviatoric shear component is the same as the total shear component. It is convenient to express the yield criterion in terms of deviatoric stress components: 3(o-i? + a£ +
(3.6)
or more concisely:
The numerical coefficients in these expressions are required because Y was chosen to be the axial stress in a plastically-deforming uni-axial tension specimen. Under these conditions: o-*1 = oru/3;
o-ii = 2on/3;
o ^ = ^33 = -an/3
(3.8)
and all the shear components are zero. The function of deviatoric stress components on the left-hand side of equation 3.7 occurs frequently in plasticity theory and it is useful to define the square root of this to be the generalised stress a:
3.2.3
Constancy of volume in plastic deformation
An important consequence of the division of the stress into deviatoric and hydrostatic components is that it separates the distortional and dilatational elements of the strain.
3.2 Elements of plasticity theory
43
Hydrostatic stress is uniform and has no effect on the conditions for yielding, though it may cause a normally brittle material to deform plastically by suppressing the tensile failure, which is a quite different matter. Because hydrostatic stress does not produce shear, there is no plastic change of shape; the effect of hydrostatic stress is to produce elastic change in volume which is fully recoverable on release of the load. A corollary of this is that the plastic deformation due to shear does not produce a change in volume. This is a well-established experimental fact, and is indeed one of the important constraints that must be imposed upon a plastic or elasticplastic FE solution. If a deformation is assumed to be purely plastic, with no elastic strain, then in terms of the principal strains e,: dV=0;
dei + de2 + de3 = 0
(3.10)
however large the individual strain components may be. This is the simplification made by rigid-plastic and visco-plastic FE formulations. As we have seen, in reality plastic deformation will be accompanied by elastic changes in volume. In the more rigorous elastic-plastic formulation that is the main subject of this monograph, the strain increments in equation 3.10 are interpreted as being the plastic parts of the total strain increments.
3.2.4.
Decomposition of incremental strain
The assumption is made that the plastic and elastic components of incremental strain can be separated. This is believed to be strictly valid for infinitesimal strains and appears to represent real behaviour accurately also for finite strains. Thus: deij = deetj + de% (3.11)
3.2.5
Generalised plastic strain
In the same way that a generalised stress was defined, a generalised plastic strain increment can be written in the form: )/2
(312)
in which it should be noted that de?y = de?j, since the volume component of plastic strain increment is zero. Once again, the coefficient in equation 3.12 is chosen so that de p = defi for simple plastic tension, since in this situation, the condition for zero plastic volume change gives that: d? and the shear components of incremental strain are again zero.
(3.13)
44
Small-deformation elastic-plastic analysis
The accumulated plastic strain £p is defined to be the sum of all the infinitesimal increments of plastic strain from the start of the deformation: (3.14)
3.2.6
The relationship of stress to strain increment: the PrandtlReuss equations
In viscous flow of a Newtonian fluid, the shear stress is proportional to the shear strain rate:
°-'7=T»-^-f
(315>
This equation, or a modified version in which the coefficient of viscosity rj is not constant, can be used in the FE analysis of metal or polymer deformation at high temperature. Many metalforming problems have been handled by this method [3.1,3.2]. An alternative approach, adopted in this chapter, is to ignore in the first instance the influence of strain rate and to concentrate on the hardening effects of strain, as in cold metalforming. According to Reuss [3.3], there is a direct proportionality between the deviatoric stress and the plastic strain-increment: dep; = a'ijdk
(3.16)
To this must be added the elastic strain:
where G is the Shear or Rigidity Modulus. The condition of volume constancy in plastic deformation is: de?, = 0
(3.18)
and for hydrostatic tension or compression (Appendix 7): E Collecting these components together: del, = deTj + 2G
(3.20a)
de,, = del + \-2v , h. do-,, These are the Prandtl-Reuss equations [3.4]
(3.20b)
3.2 Elements of plasticity theory
3.2.7
45
Elastic-plastic constitutive relationship
We require for an elastic-plastic FE formulation a relationship between the incremental strain and the incremental stress in a plastically-deforming body similar to equation 2.17. The starting point is the Prandtl-Reuss equations 3.20. In Appendix 3 it is shown how these can be used to derive a relationship of the form:
dot, = 2G ( de, + 8, ( - ^ - )
)
(3.21)
where:
and Y' is the rate of change of yield stress Y with respect to plastic strain e p . In terms of the stress increment vector: do- = (dcr n , d a ^ , dcr33, do"12, da23, do-13)T
(3.23)
and the strain increment vector: de = (den, de 22 , de 33 , dy 12 , dy 23 , dy 13 ) T
(3.24)
this relationship may be expressed as: do- = [D]de
(3.25)
where: [D] = [D<] - [D>] e
(3.26) p
[D ] is the elastic stress/strain matrix presented in Chapter 2; [D ] is described in Appendix 3.
3.2.8
Strain hardening in FE solutions
For small but finite increments of deformation, we may generalise equation 3.25: A*=
[D]Ae
(3.27)
For an infinitesimal strain increment, the value of the slope Y' in equation 3.22 is found from the tangent at the appropriate strain. The slope, however, generally decreases with strain, so that at the end of a finite increment of deformation the value will be lower. There are several ways of improving the accuracy of the constitutive relationship for finite deformation steps. The simplest of these uses the mean value of the tangents at the beginning and end of the step, which is essentially the same as using the slope of the secant, as in figure 3.2. This is therefore referred to as the secant modulus method. Other approaches will be discussed later.
Small-deformation elastic-plastic analysis
46
tangent secant
Fig. 3.2 The secant approximation to slope Y' for a finite deformation.
3.2.9
Force/displacement relationship: the stiffness matrix
We assume a constant stiffness over each small increment of deformation and obtain a relationship between incremental displacement Adi and incremental force Afi for element /: [tfi]Ad,= Afi (3.28) The element stiffness matrix takes the same form as that derived in Chapter 2 (equation 2.36): (3.29) but now we use the elastic-plastic constitutive matrix associated with the current state of element / (q.v. equation 3.26).
3.3
EXAMPLE ANALYSIS USING THE SMALLDEFORMATION FORMULATION
To illustrate the use of the small-deformation formulation, we return to the simple example of a tensile test considered in Chapter 1. Figure 3.3 shows the FE mesh used in this example. As before, the mesh represents an axi-symmetric test specimen with gauge diameter 11.2 mm and gauge length 100 mm. Nodal points 7 and 8 are assumed to be fixed and prescribed axial displacements of 0.25 mm per increment are imposed upon nodes 1 and 2. The value of E is 2 x 105 N/mm2 and v is 0.33. The material yields initially at a stress of 80 N/mm2 and the slope Y' is assumed to have constant value of 100 N/mm2. The deformation is modelled using the BASIC demonstration program listed in Appendix 11, and is continued for 10 increments - corresponding to 5% elongation. By this stage all the elements have started to deform plastically, and
3.3 Example analysis
47
a neck at the mid-point (node 2) is clearly beginning to form (figure 3.4). The very coarse mesh results in the neck being far more diffuse than would be the case in practice, with large plastic deformation taking place at some distance away from the mid-point. This example is, however, a useful demonstration of the technique. Fig. 3.3 (a) tensile-test specimen and (b) FE model, (a)
(b)
*1 Fig. 3.4 Simulated deformation and distribution of generalised stress (N/mm2) for tensile test. 89
135b
126
170
165
^ A 154 j | original profile 112
48
Small-deformation elastic-plastic analysis
References [3.1] Rebelo, N. and Kobayashi, S. A coupled analysis of visco-plastic deformation and heat transfer. Int. J. Mech. Sci. 22, 699-705, 707-18 (1980). [3.2] Pittman, J.F.T., Zienkiewicz, O.C., Wood, R.D. and Alexander, J.M. (Eds.) Numerical Analysis of Forming Processes, Wiley (1984). [3.3] Reuss, A., Beriicksichtigung der elastichen Formanderung in der Plastizitatstheorie. Zeits angew. Math. Mech. 10, 266-74 (1930). [3.4] Hill, R. The Mathematical Theory of Plasticity, Clarendon Press (1950).
4
Finite-element plasticity on microcomputers
4.1
MICROCOMPUTERS IN ENGINEERING
The previous chapters of this book have focussed attention on the basic FE theory for elastic and small-deformation elastic-plastic applications. The latter were illustrated by a very simple analysis of the tensile test. In this chapter, we shall look at problems more closely associated with realistic metalforming processes. The first part of the chapter examines a PC-program written in BASIC that can be used for demonstrations or for student tutoring. A listing of this program is given in Appendix 11. The second part of the chapter describes a FORTRAN program, implemented on a large-memory micro, that is designed for more serious metalforming studies. No attention has been given so far to the machines on which FE analyses can be performed. Where mainframe computers or workstations are available, this is not important but in microcomputer applications, how the machine operates and the importance of selecting the right operating system, software and processors cannot be taken for granted. The decision as to what equipment is most appropriate must be made by the individual user, based on carefully assessed requirements. A general introduction to microcomputers is given by Avison [4.1] in which many aspects of micro hardware and software are discussed, and comment on buying a micro is included. Hardware and software selection is also discussed by Samish [4.2] and Lane [4.3] respectively, while Bell [4.4] gives an overview of the role of computers in engineering. Some background knowledge is essential when trying to identify the merits or otherwise of various microcomputer systems. When the various single-user or multi-user operating systems are considered, together with different programming languages, central processors and also peripherals, the task of selecting the right system appears formidable. But with
50
Finite-element plasticity on microcomputers
reference to the predetermined requirements a short list of suitable combinations can soon be produced from which the final selection can be made. The use of micros to aid the solution to problems in many fields of engineering is rapidly expanding. These cover heat transfer, piping design, solids handling and transport modelling as well as the usual CAD and general stress analysis systems. O'Connell etal. [4.5] list 235 packages solely for mechanical engineering design. These commercial packages however, represent only the beginnings of a comprehensive software range for engineers. Most of the software developed to date is for linear problems and once the appropriate algorithms have been established their implementation on a micro is not unduly demanding. The major problems, whether on a mainframe or a micro, are in the initial specification of the problem and in the interpretation of the mass of results that computers can quite easily produce. The solution of non-linear problems on micros introduces additional complexities. Either the random access memory (RAM) of the machine is too small or the running time is too long, or both. Careful attention must be given to how the program is structured to take full and efficient advantage of the micro. Killingbeck [4.6] describes many features of developing programs for scientific applications in physics. Brown [4.7] lists and describes the construction of various FE programs written in BASIC for solving mechanical engineering problems on a micro. In contrast to the linear analysis systems, very few non-linear programs have been written. Examples include time-dependent heat flow problems [4.8, 4.9] and the dynamic behaviour of structures [4.10]. The FE analysis of non-linear plasticity problems poses two major obstacles to its implementation on a micro. One is the large amount of data that needs to be handled and repetitively transferred to and from storage, and the second is the need to perform an analysis in numerous increments, each at least equivalent to a linear elasticity problem. The following sections describe how these obstacles may be overcome to solve realistic metalforming problems. All the examples in the following sections of this chapter are based on smalldeformation plasticity theory, as described in the previous chapter. The more correct, and more useful, large-deformation plasticity theory on which any critical analyses should be based is described in subsequent chapters.
4.2
NON-LINEAR PLASTICITY DEMONSTRATION PROGRAMS ON A MICROCOMPUTER USING BASIC
4.2.1
Introduction
Innumerable variations in microcomputers and operating systems together with the wide variety of peripherals, result in each particular combination having its
4.2 Demonstration programs using BASIC
51
own particular characteristics. The interchangeability of components and compatibility of microcomputers is still far from ideal. The introduction of non-linear programs to microcomputers is a demanding task on both developer and machine, and any system will require some modification when a program is transferred from one machine to another. Despite this, there are a number of techniques which can be used to aid the development of non-linear programs.The techniques themselves will be applicable to any micro system though of course their implementation will vary to suit the specific hardware and software requirements. This section will describe techniques used for running a non-linear program on small (in terms of RAM capacity) machines. These programs are particularly suitable for demonstration or tuition purposes.Techniques for much larger micros for more realistic simulation will be described in Section 4.3.
4.2.2
Structure of the finite-element program in BASIC
The small micro used for the example here is an Olivetti M24, but similar programs have been run also on Sirius, IBM,Tektronix and BBC microcomputers, with minor changes in the file handling and graphics. A small-strain FE formulation, described in Chapter 3, is used and the program to examine 2-D metalforming problems is written in BASIC. This in itself does not pose any difficulties. The greatest problems arise in trying to structure the FE program into a form which could be run using the small amount of RAM available. A listing of the program developed is given in Appendix 11. A simple flow chart for a mainframe program is shown in figure 4.1 and depicts the familiar sequence of calculating the [£j] and [Dj] matrices, followed by the element stiffness matrices [Ki\. Once the global stiffness matrix [K] is complete, the displacements, stresses and strains are found as usual. Each of these operations can easily be accommodated within a mainframe computer memory, but for the smaller machine this is not the case and frequent use of disc storage must be made. The new structure required to accomplish this is illustrated in the flow chart shown in figure 4.2. The BASIC FE program consists of a series of separate files on a disc. One file contains all the instructions necessary to set up the computer model. This initial generation of the data may be accomplished with a small predetermined program or interactively. In either case the data must be transferred to a specified file on the disc. The initial program is then deleted from RAM (although retained on disc) and the next file loaded. This contains the routines to construct the [Bi] and [Dj] matrices for each element. This file must therefore recall the relevant mesh and property data from disc storage, evaluate the above matrices, then store this information back on the disc. This is continued until all the elements in the mesh have been considered, and the RAM is then cleared. The program is directed to the next file and, after retrieving the data previously set up for
Finite-element plasticity on microcomputers
52
each element in turn, evaluates the element stiffness matrices [KJ, and finally the global stiffness [K]. Once again this information is transferred back to the disc and the RAM cleared. The final file is then loaded for retrieval of the stiffness matrix. The displacement of each nodal point is evaluated, and hence the strain and stress in each element may be determined. The material properties and mesh geometry are then updated ready for the next increment. Although this approach has been described as one of interchange between machine and disc, at no time is any of the program detail deleted from the disc.
4.2.3
Simple application and comparison to mainframe results
The familiar process of simple axi-symmetric upsetting is chosen in order to examine the operation of the BASIC program. The workpiece and FE mesh are shown in figure 4.3. The mesh is very crude but is sufficient to demonstrate the micro program. Fig. 4.1 Simplified flow chart of FORTRAN FE program used on a mainframe computer.
Read data from datafile describing FE model |
Evaluate [Bx] and [D{] matrices for each element I Form stiffness matrix [K{] for each element and add to global stiffness matrix [K] [Evaluate nodal point displacements! Evaluate strain and stress for each element I Update material properties | Update mesh geometry | Output results to filestore
Next \ ^ yes increment ^ 9
4.2 Demonstration programs using BASIC
53
In this demonstration, only the two extreme boundary conditions of zero and sticking friction are considered. The process is modelled until a reduction of height of 20% is reached. Fig. 4.2 Flow chart for BASIC FE program used on a microcomputer.
Load FILDATA and create FE model
Load FILBDM* and determine [B{] and [D{] matrices for every element
—
*
Store data in INTLDATA
y —
Store data in MATDATA
y
Load FILSTFF and evaluate [K^] for each element. Add to global matrix [ K ]
Store data in STFFDATA
y Load FILDSTS and evaluate nodal point displacements
—>-
Determine stresses and strains for each element
—
Store displacements in DISDATA
Store stresses in STSDATA
Load DISPLAY and display illustrations of metal flow
main program route data storage data retrieval *FILBDME for elastic analysis FILBDMP for elastic-plastic analysis
54
Finite-element plasticity on microcomputers
applied displacements
8 TRI3 elements 9 nodes 18d.o.f.
billet
J
Fig. 4.3 Cylindrical billet and FE model.
In the first case, with theoretically no surface restraint, the workpiece should deform in a completely uniform manner. As a further check the uniform plastic strain that should develop in the workpiece can be evaluated easily from ep = \n(H/h), where H is the initial height and h is the final height of the billet. In this case, at 20% reduction, ep = 0.22314. Figure 4.4a shows the FE predictions using the BASIC program. The model has clearly deformed homogeneously with a generalised plastic strain of 0.2209, an error of 1%. Fig. 4.4 Distorted grids and distribution of generalised plastic strain for (a) zero friction and (b) high friction at 20% reduction in height. Results using Olivetti microcomputer. original profile 1 1 0 /
0.2209> //
2209 / > ' 0.2209
> / 0.2209
o.:>209
0.2209
>/
/
/'o
y
v^O.2209'
2209 (a)
(b)
4.2 Demonstration programs using BASIC
55
When the second limiting boundary condition is imposed, no relative movement of the die/billet interface occurs, and the deformation pattern is very different. The high interface friction results in a pattern of flow in which a central zone with little plastic deformation ep = 0.1095, is formed in the area in contact with the die. This accords with the known pattern of a conical dead zone, as suggested also by the plane-strain slip-line field analogy. The majority of plastic deformation, according to slip-line theory and experiment is restricted to diagonal regions located across the billet joining opposing corners on the section. This can be seen in broad terms in the simple model. Intermediate levels of deformation are found near the equatorial free surface. All these features, even in restricted detail due to the coarse mesh, can be identified on the FE predictions using the micro and illustrated in figure 4.4b. Analyses with identical meshes, and another with a much finer mesh [4.11] have also been performed using a CDC7600 mainframe computer for which the programs were initially developed. In the case of zero friction the average plastic strain for the mainframe coarse mesh analysis is 0.222, only 0.5% less than the theoretically-exact value. There are some small variations found in the distribution of plastic strain (figure 4.5a). The predictions using the micro show less variation. The mainframe solution for sticking friction, figure 4.5b, gives a pattern of deformation similar to that found with the micro. As a further check on the results from the micro, the variation of plastic strain along a circumferential section may be compared to that obtained using a much finer mesh of 842 elements on the mainframe. In general good agreement is found (figure 4.6). These results clearly demonstrate the feasibility of adapting non-linear programs to run on small machines. These provide valuable demonstration and Fig. 4.5 Distorted grids and distribution of generalised strain for (a) zero friction and (b) high friction at 20% reduction in height. Results using a CDC7600 mainframe computer. original profile
(a)
Finite-element plasticity on microcomputers
56
tuition examples, allowing a student to observe the construction of the program in full detail and even to check selected calculations manually. The programs can be fully interactive and give a good background to real, practical, analysis.
4.3 4.3.1
A 'LARGE' FORTRAN-BASED SYSTEM FOR NON-LINEAR FINITE-ELEMENT PLASTICITY Hardware selection
The following features are suggested as desirable requirements for the microcomputer system: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)
16-bit machine Programming language should be FORTRAN Operating system should be standard Large RAM should be available Disc drive, preferably one floppy and one hard disc Single user system Good resolution graphics System should include printer and plotter
The 16-bit machines offers some advantages over 8-bit machines in terms of accuracy and speed, and the more powerful 32-bit models can be recommended if funds permit. The use of FORTRAN as the programming language not only aids future transfer of the program from one micro to another but makes the task of transference from a mainframe computer easier. The machine RAM must be at least 256k with simple facilities for additional memory. The availability of a hard disc drive allows the large amounts of data produced by non-linear FE analyses to be easily and rapidly stored ready for further processing if necessary. Graphical display of the results is particularly important. A reasonable screen resolution is important for assessing the output. Fig. 4.6 Generalised plastic strain predictions using various computers and meshes for the circumferential section AB: CDC7600, 842 TRI3 elements; -— CDC7600, 8 elements; — Olivetti M24, 8 elements.
'B
0 0.25 (^4) axial position
0.75 xjh
1.0 (B)
4.3 A large' FORTRAN-based system
57
Provisions must be made for producing hard copies of the results, so any system should be complemented with a suitable printer and plotter. Numerous 16-bit machines are available and though the majority are developed for business use, not scientific applications, several can, in principle, be used. The final choice will involve reliability, after-sales support, country of manufacture, delivery time, location of suppliers and the total cost. A number of systems would meet the requirements but the following system was set up for the work described here: Microcomputer Operating system Main processor Working RAM Programming language Disc storage Graphics Peripherals
Future Computer FX30 Concurrent CPM-86 Intel 8088 (8 MHz) 768k SSS FORTRAN 784k floppy, 10M hard disc PLUTO graphics system Dot matrix printer, A4 plotter
The concurrent CPM-86 operating system was included to enable more than one task to be undertaken simultaneously. This was particularly useful for program development when frequent changes were made. The programming language, 'Small Systems Services' FORTRAN (SSS FORTRAN) is equivalent to ANSI-66 FORTRAN. The normal directly addressable space in RAM for a 16-bit machine of the type used here is 64k. The additional RAM could be accessed by using specially prepared routines written in assembler language or by using a different FORTRAN compiler as described in section 4.3.5. One floppy disc was augmented with a 10 Mbyte Winchester disc. An A4 flat bed plotter and a dot matrix printer were also included. The components of the system are illustrated schematically in figure 4.7, and further details can be found in reference [4.12]. Fig. 4.7 Components of the microcomputer system hardware and schematic layout. Switch
A4 plotter
VDU
FX3Q processor
Keyboard
[-
I
Plotter
Graphics VDU
j
Graphics processor
|
58
Finite-element plasticity on microcomputers
4.3.2
Program transfer
Two small-strain programs were transferred from the mainframe computers. These were the programs developed by Hartley [4.11] and Al-Sened [4.13] for the study of 2-D metalforming problems. The former used only constant-strain triangular elements with three nodes (TRI3), while the latter was extended to use eight-node isoparametric quadrilateral elements (QUAD8).The major contents of the programs remained unaltered but the means of data input/output and graphics all require modification. The SSS FORTRAN compiler requires a minimum of 128k of RAM for the initial program compilation. The operating system requires 40k and the executable file of either program needs 96k. So for running any sensible problem within the machine RAM, thus avoiding many data transfers, at least 200k is necessary to allow for reasonable amounts of data storage. Constants and blank common blocks are limited to 64k blocks. This imposes some restriction on the number of elements that could be considered in a model if the program is to run entirely within the machine RAM. Although this restriction can be overcome by various methods the examples in the immediately following sections are all conducted within the above limitations. Later sections (4.3.4 and 4.3.5) describe the alternative methods and further examples are given.
4.3.3
Applications of the non-linear system
4.3.3.1
Axi-symmetric upsetting
Simple upsetting of an axi-symmetric billet between flat parallel platens is used here for initial assessment of the microcomputer programs. The process was analysed using bothTRB elements and QUAD8 elements. The results are compared with those obtained using the original mainframe programs. Previous experimental results and solutions using the mainframe FE programs [4.14] are available. For the present analysis high friction (no lubricant, m = 0.7) is assumed at the die and platen interface. (See Chapter 6 for a description of the technique used to include the effects of interface friction.) The FE mesh representing the aluminium workpiece is shown in figure 4.8. The internal deformation predicted by the microcomputer program usingTRB elements is shown in figure 4.9. It can be seen that the profiles clearly indicate significant barrelling. The edge profiles exhibit some oscillations at high reductions in height, attributable to the coarse mesh. A finer mesh would give more precise results. The predictions are almost identical to those produced by the mainframe program. Figure 4.10 shows the distribution of generalised stress values, at 50% reduction in height of m = 0.7. The average percentage variation in the stress values compared to those produced by the mainframe program is 0.06%.
4.3 A 'large' FORTRAN-based system
59
The computing times for these examples reveal an interesting trend. Figure 4.11 shows the computing time plotted as a function of the total number of degrees of freedom for the two solutions corresponding to the use of (a) the mainframe computer and (b) the microcomputer. The percentage increase in computing time with the number of degrees of freedom is roughly the same for both computers, but the micro takes about 40 times as long for this type of solution. The predictions for simple upsetting with the QUAD8 elements are closely similar to those obtained withTRD elements. In general fewer of these elements Fig. 4.8 Initial FE meshes for upsetting. arrangements of elements
QUAD8
TRI3
Fig. 4.9 Predicted grid distortion in simple upsetting using a microcomputer withTRB elements at 50% reduction with high friction.
Fig. 4.10 Generalised stress (N/mm2) distribution predicted with TRI3 elements at 50% reduction using the microcomputer. 160
Finite-element plasticity on microcomputers
60
o
100
150
300
200
degrees of freedom
Fig. 4.11 Computing time per increment for (a) the mainframe computer and (b) the microcomputer. are needed to achieve the same degree of accuracy. Although the use of QUAD8 elements does not reduce the computational time, the reduction in the number of elements generally reduces the file space needed to contain the data and results. 4.3.3.2
Cold heading
To assess the use of both types of element in a more complicated process, cold heading of an axi-symmetric billet is considered (figure 4.12). The specimen has a diameter of 50 mm, a free-length to diameter ratio of 1 and a gripped length of 50 mm. The incremental displacement of the upper punch for each step is 1% of the original free height of the billet. Figure 4.12 shows the initial FE meshes. Fig. 4.12 Initial FE meshes for cold heading. flat punch / / / / / / / / / / / / / / QUAD8
gnpper dies
//
s///
. TRI3
-billet
4.3 A 'large' FORTRAN-based system
61
Symmetry of the die about the centre line of the billet means that the solution need only be carried out over half the billet. This symmetry determines the boundary conditions on the centre line, where shear traction and normal displacement are both zero. Two models are used for the theoretical comparisons. In the mesh used for the first model, 220TRI3 elements with 254 degrees of freedom and 220 integration points are used. In the second model, 30 QUAD8 elements with 224 degrees of freedom and 120 integration points are used. Again, m = 0.7 is used for high friction. The effect of friction in the gripper dies is not considered. The distorted grids are shown in figure 4.13. The distorted grid lines are somewhat smoother with the QUAD8 elements than with the TRI3 elements and in particular the local deformation at the inner corner below the head is more accurately modelled by the less rigid QUAD8 elements.
4.3.4
Overcoming the FORTRAN compiler limitations
To allow more elements to be used in the analysis it is necessary to overcome the restrictions on the amount of normally addressable RAM that can be accessed. The major problem in the non-linear analysis is storing the banded stiffness matrix during the solution routine. The direct Gaussian elimination technique has been retained throughout this work (see Appendix 5). Afront solver technique would reduce the storage required but would also result in increased computing times. When not located in the RAM a large stiffness matrix can be stored in one of two ways: (i) The matrix can be sorted in the out-of-bounds memory within the machine memory but not within the current RAM being used for the main program. This is designated EPFEPAOM Fig. 4.13 Predicted grid distortion in cold heading at 50% reduction and high friction, (a) using QUAD8 elements and (b) usingTRB elements. Microcomputer results. V \
L-fN
(a)
OsW
4-
\
Y/
(b)
\
X
62
Finite-element plasticity on microcomputers
(ii) The matrix can be stored in a random access file on the hard disc. This is designated EPFEPMSD These two methods together with the original version (EPFEPM) are compared in terms of computing time for various numbers of elements in the FE mesh. Once again the familiar process of simple upsetting with high friction will be used as the benchmark test. Only QUAD8 elements will be considered. In the original version EPFEPM, a maximum of 30 elements could be used. This could be increased to about 250 for the EPFEPAOM technique and quite easily 1000 elements when the EPFEPMSD technique is used. Figure 4.14 shows clearly the expected result that the programs run more quickly if data transfer to any medium can be avoided. When the RAM storage limit is reached data transfer to the hard disc is the most efficient.
4.3.5
Introducing an improved FORTRAN compiler and the 8087 maths co-processor
4.3.5.1
System improvements
The system developed allows the analysis of many practical plane-strain and axi-symmetric metalforming problems; for many 2-D processes 100 QUAD8 elements are adequate except where severe geometric changes are involved. To reduce running times to an acceptable level for these, two further enhancements to the system can be introduced: (i) the addition of an 8087 mathematics co-processor (ii) the addition of a new FORTRAN compiler, Pro-FORTRAN The 8087 maths co-processor is specifically designed to deal with mathematics operations and does so much quicker than the 8088 processor. It does not replace Fig. 4.14 Computing times per increment for three different data storage and transfer techniques. Microcomputer results. 10 EPFEPAOM
EPFEPMSD
EPFEPM 50 100 150 number of QUAD8 elements
200
4.3 A 'large' FORTRAN-based system
63
the latter chip but operates alongside it. To make full use of the maths processor the appropriate library routines in the FORTRAN compiler must be specified. Although not done here some programs may also need re-structuring to speed up the mathematics processing [4.15]. The Pro-FORTRAN compiler allows the RAM memory to be accessed in 64k pages. This access to large blocks of data will allow the data transfer to be done much quicker than using the EPFEPAOM method described earlier which could transfer data only in very small units. The paging type of access is achieved by incorporating additional routines written in assembler language within the FORTRAN compiler. Incorporating the 8087 mathematics co-processor reduced the running times of the examples shown earlier to about 70% of the original processing time.The Pro-FORTRAN compiler reduced this by a further 75%, so the running times were typically a fifth of the original. This reduction was largely due to the reduced time in data transfer in RAM locations. 4.3.5.2
Analysis of upsetting and heading with refined meshes
To demonstrate typical results that can be achieved with realistic FE models on a micro, two examples, upsetting and heading, are again considered. The upsetting process is analysed using 440TRI3 elements, and distorted grids and displacement vectors are shown in figure 4.15. These are very similar to the earlier results but show improved free surface profiles and more internal detail. The running time for this mesh was 1850 seconds per increment. Fig. 4.15 Microcomputer predicted distorted grids and displacement vectors in simple upsetting.
I I
77//// / i
N,
N\
S
S
N
V
I \
I i
\ l
1 \
1 \
\
\ \ \ \ \
*\ \ \ \
I I
if///
I
/
I
/
1
.
t
.
t
/
64
Finite-element plasticity on microcomputers
Heading is analysed using 105 QUAD8 elements. Figure 4.16 shows improved results compared to those found earlier. The running time in this case was 1440 seconds per increment. The introduction of the 8087 co-processor and the Pro-FORTRAN compiler means that realistic metalforming analyses can be conducted within 24 hours on a microcomputer. This may still seem lengthy to the user who, having seen trivial results produced immediately on a VDU, expects the computer to produce all results instantly. Nevertheless, the present system offers a practical means of obtaining very detailed solutions to metalforming problems relatively quickly. 4.4
SUMMARY
In this chapter we have seen examples of metalforming process simulation using a small-strain FE formulation. This type of formulation is also referred to as 'small-deformation' or as infinitesimal-strain'. The main point here is that this approach is strictly applicable only to situations in which the plastic strain is of the same order of magnitude as the elastic strain. When this is not the case, the small-strain formulation cannot calculate accurate values of the components of strain and stress, even though, as shown by the upsetting and heading examples in this chapter, the generalised values of strain and stress and the overall pattern of flow may be reasonable. For a complete and much more accurate model of metalforming plasticity, a more rigorous formulation, the finite-strain approach, must be used. This approach is described in the following chapters.
Fig. 4.16 Microcomputer predicted distorted grids and displacement vectors in cold heading. \ \ \ \ x
References
65
References [4.1] Avison, D.E. Microcomputers and their Commercial Applications, Blackwell Scientific Press (1983). [4.2] Samish, F. Choosing a Microcomputer, Granada (1983). [4.3] Lane, J.E. Choosing Programs for Microcomputers, National Computer Centre (1980). [4.4] Bell, W.T. The role of microcomputers in civil and structural engineering. Engineering Software II, Proc. 2nd Int. Conf on Engineering Software, ed. R.A. Adey, CML Pubs., pp. 852-61 (1981). [4.5] O'Connell, C , Sheviak, J.K., Browne, A.R. and Johnson, S.V. Directory of Microcomputer Software for Mechanical Engineering Design, Marcel Dekker (1985). [4.6] Killingbeck, J.P. Microcomputer calculations in physics. Rep. Prog. Phys. 48, 54-99 (1985). [4.7] Brown, D.K. An Introduction to the Finite Element Method using Basic Programs, Surrey University Press (1984). [4.8] Moir, P. J. and Stoker, J.R. Theta: A desktop computer solution for nonlinear time dependent heat flow problems. Engineering Software II, Proc. 2nd Int. Conf. on Engineering Software, ed. R.A. Adey, CML Pubs., pp. 796-805 (1981). [4.9] Stelzer, J.F. Consideration and strategies in developing finite element software for desktop computers. Eng. Comp. 1, 106-24 (1984). [4.10] Backx, E. and Rammant, J.P. Structural dynamic interactive analysis in Basic on micros. Engineering Software II, Proc. 2nd Int. Conf. on Engineering Software, ed. R.A. Adey, CML Pubs., pp. 842-51 (1981). [4.11] Hartley P. Metal flow and homogeneity in extrusion-forging, Ph.D. thesis, University of Birmingham, UK (1979) (unpublished). [4.12] Hussin, A. A.M. Transference of mainframe finite-element elastic-plastic analysis to microcomputers, and its application to forging and extrusion, Ph.D. thesis, University of Birmingham, UK (1986) (unpublished). [4.13] Al-Sened, A. A.K. Development of simulation techniques for cold forging sequences, Ph.D. thesis, University of Birmingham, UK (1984) (unpublished). [4.14] Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Influence of friction on the prediction of forces, pressure distributions and properties in upset forging. Int. J. Mech. Sci. 22, 743-53 (1980). [4.15] Startz, R. 8087 Applications and Programming for the IBM PC and other PCs, R.J. Brady Company, Prentice-Hall (1983).
5
Finite-strain formulation for metalforming analysis
5.1
INTRODUCTION
The first applications of the FEM were concerned with structural problems, and so it is not surprising that when the FEM began to be applied to the modelling of plastic deformation, this was viewed simply as an extension of non-linear elastic behaviour, with the elastic stress/strain matrix in the FE formulation replaced by an appropriate plastic one, as described in Chapter 3. When the amounts of plastic deformation and material displacement are small, this is a valid approach and is often used in the study of plastic failure of structures such as pressure vessels. The small-strain technique also has the advantage of being easily understood, thus providing a good introduction to the principles and underlying FE plasticity, and of being easily incorporated into available elastic FE programs. As the examples in the previous chapter show, the small-strain approach can give good approximations to the overall pattern of deformation in certain simple forming processes. However, this technique cannot accurately predict the values of important workpiece parameters, such as the components of strain and stress, particularly if the metalforming process involves appreciable material rotation. This is due partly to the nature of elastic-plastic deformation, for which correct definitions of stress and strain increment must be chosen, and partly to the large total and incremental strains involved in metalforming, which require special numerical techniques for their evaluation. This chapter re-examines the FEM in the light of these considerations in order to derive a formulation that is able to model large-strain elastic-plastic deformation. The next chapter considers how this formulation can be implemented in a computer program to study practical metalforming operations. The theory so far has been presented mainly in matrix notation, since this is the most familiar approach. However, the ideas that are considered in this chapter
5.2 Governing rate equations
67
are more clearly, and more naturally, expressed using tensors. For those unfamiliar with suffix notation, or with the concept of tensors, Appendix 8 gives a brief introduction to the subject. The FE formulation is derived here for a general 3-D deformation: the corresponding 2-D expressions (for plane-strain, plane-stress or axi-symmetric geometries) may be obtained, if necessary, as a special case of the 3-D ones. It should be noted that throughout this chapter and the next, all expressions refer to a single element of the FE mesh. However, for reasons of clarity, the subscripts denoting this have been omitted.
5.2
GOVERNING RATE EQUATIONS
5.2.1
Rate of potential energy for updated-Lagrangian increment
Since metalforming processes may involve large amounts of deformation, the FE analysis must be divided into a number of steps, or increments, each considering just a small part of the total deformation. Consider an element of the body at the start of an increment of deformation. Let the reference Cartesian co-ordinates of a particle at point P within the element be xt. During the increment, the element deforms so that at some later instant the reference co-ordinates of the particle, now at point P', are x\. These co-ordinates will be functions of the initial co-ordinates and of time. The total rate of increase in potential energy of an element of the deforming material is the rate of increase in the internal deformation energy of the material, less the rate of decrease of the potential energy due to the movement of the points of application of external applied forces. The average rate of increase of the internal energy per unit volume is the dyadic product (that is, the tensor equivalent of the scalar product of vectors) of the deformation-rate tensor and the average stress tensor. In the updated-Lagrangian approach considered here, the state variables are specified with respect to the reference configuration at the start of the increment, so the rate of change of internal energy is required per unit initial volume. The deformation rate is therefore the derivative of the instantaneous velocity of a particle with respect to its initial reference co-ordinates. Thus, if the displacement of the particle is ut = x\ -xh the deformation rate is: u,j=—
dXj
(5-1)
For the same reason, the correct stress to use in this context is the nominal stress Sij (see Appendix 9 for a description of the different stress tensors). The FEM assumes that the forces acting on the body are concentrated at the nodes of an element. If fIm is the rath reference component of force acting at
68
Finite-strain formulation
node /, and dIm is the corresponding component of the displacement of this node from some convenient starting point, the average rate of increase of potential energy / of the element during time dt is: .
/ =
r
1
1.
Jtea+ y ^ d O M V - ( / > m + y//md*Hm
(5.2)
where the integration is carried out over the volume Vbf the element. Henceforth, the usual summation convention will be extended to include summation over all the nodes of an element when an upper-case subscript is repeated in a term of an expression.
5.2.2
Minimisation of rate of potential energy
The velocity field is such that the rate of increase of potential energy of the element is a minimum. By the calculus of variations, the condition for the functional / to have a stationary value with respect to the functions u]-ti and dIm is that: S(i) = i(slj+Yiiidt)S(ujJ)dV-(flm
+ ^-flmdt)d(dIm) = O
(5.3)
in which 8() denotes an arbitrary variation in the function enclosed in the parentheses. Since the element is in equilibrium at the start of the increment, the first-order terms in equation 5.3 cancel to give:
5.2.3
Incorporation of strain rate
To complete the governing rate equations, it is necessary to express the stress rate in terms of strain rate by means of the appropriate elastic-plastic constitutive law. The strain rate is defined to be the symmetric part of the deformation-rate tensor: 1 £ij= y(w/,/+My,/) (5.5) The constitutive relationship normally involves the rate of true or Cauchy stress o-y, but since the integration in equation 5.4 is carried out over the initial volume of the element, the correct measure to use in this instance is the rate of Kirchhoff stress r7y, which is simply equal to the Cauchy stress multiplied by the ratio of the current to the initial volume. Furthermore, the rate of stress used in the constitutive relationship must vanish during rigid-body rotation, when e,-/ vanishes. For this reason, the time derivative of Kirchhoff stress is measured in the Cartesian co-ordinate system, which is co-incident with the reference system at the start of the increment, and which rotates with the element. This derivative
5.2 Governing rate equations
69
is often called the rotationally-invariant or Jaumann derivative f*y. In Appendix 10, it is shown that this is related to the rate of nominal stress by: Sij = T*y - (Tkjkik - (Tikkkj + <JikUjjk
(5.6)
The elastic-plastic constitutive law therefore takes the form: rlj = Dljklekl (5.7) Substitution of equations 5.6 and 5.7 into equation 5.4, using the symmetric properties of the Kirchhoff stress-rate and the strain-rate tensors, gives the basic rate expression: Kdim)hm = l[s{iii)(Dijkl
- 2<Tik8$kkl + 8{ujti) aikuhk]dV
(5.8)
where fy is the Kronecker delta introduced in Chapter 3.
5.2.4
Importance of correct choice of stress rate
In FE formulations designed for small-scale plasticity, the constitutive relationship is assumed to be between strain rate and the rate of nominal stress. This leads to a much simpler FE governing expression but is invalid for formulations intended for metalforming applications, in which gross deformation can occur. It has been shown [5.1] that the use of the nominal stress rate in the flow expression leads to significant errors in the resulting stiffness equations unless the current modulus of plasticity is much greater than the current yield stress, a situation rarely met in practical metalforming processes. The small-strain technique, therefore, cannot accurately model plastic deformation however small the increments are made. In addition, a small-strain formulation may be inaccurate, even for small-strain elastic deformation, if the magnitudes of the displacements are large, as the following example shows. Two FE analyses of the elastic deformation of a single cubic element are performed (figure 5.1), in which simple tension is combined with rotation. In one of the FE analyses, the small-strain constitutive relationship, involving nominal stress rate, is used while, in the other, the formulation is based upon the finite-strain governing equation 5.8. The element is assumed to start with a tensile stress in the x x direction; the stress and forces acting on the element at any later stage in the rotated co-ordinate system can easily be deduced from Hooke's law. The reference components may therefore be calculated using Mohr's circle and compared with the FE results. Even when the incremental tensile strain is as small as 0.0016%, and the angle of rotation is as small as one degree per increment, figure 5.2 shows that the small-strain approach predicts incorrect values of stress and force in the element, whereas the- predictions of the finite-strain solution agree with the analytical values, even up to 90 degrees rotation.
70
Finite-strain formulation
5.3
GOVERNING INCREMENTAL EQUATIONS
5.3.1
Modification of rate expression
In the present formulation, the fundamental variables are the changes in the various parameters during each increment of the deformation, so the various time derivatives in equation 5.8 need to be replaced by the corresponding increment, during time step At, using: dImAt —
Adln
(5.9)
fImAt - AfIm
(5.10)
Fig. 5.1 Combined extension and rotation of a single element.
rotation
5.3 Governing incremental equations
f
71
-AM,,= ^ - ^ = ^ - ^
(5.11)
e,7Af- Aetj
(5.12)
where Aw/;7 is called the deformation gradient. The approximations in equations 5.9-5.12 will be valid provided the deformation increment is not too large (for example, smaller than 2 or 3%, which is larger than the value usually chosen for FE analysis).
5.3.2
Effect of rotation
By multiplying equation 5.5 by the time increment and substituting from equation 5.11, it is possible to express the increment of strain as the symmetric part of the deformation gradient. However, if the change in strain is defined in this way Fig. 5.2 Comparison of analytical values of stress and force with FE predictions using small-strain and finite-strain formulations: theoretical; O FE - small-strain; D FE - finite-strain. 100-
-100 -
-200 -
-300 -
0
10
20
30
40
50
60
70
80
90
72
Finite-strain formulation
(the infinitesimal definition) it is easily demonstrated that a non-zero value is calculated when the material is undergoing rigid-body rotation, a situation for which the strain increment should, by definition, be zero. More generally, whenever there is a rotational component to the deformation, the infinitesimal definition of incremental strain predicts an anomalous change in volume. Since in metal deformation the volume can change only elastically, the erroneous volume strain will result in the calculation of very large elastic values of hydrostatic stress.
5.3.3
LCR expression for strain increment
To avoid these problems, an alternative method of calculating the strain increment is adopted, which incorporates a correction for any rotational component of the deformation. This leads to the definition of the linearised co-rotational (LCR) strain increment [5.2]: 1 f 1 Ae l 7 = — I rkiAukJ + rkjAukJ-—(
1 + k>) (5.13)
where the angle brackets denote the skew-symmetric part of the enclosed tensor, for example: =y(A« /J -A U/ ,,.)
(5-14)
rik is the rotational part of the unique decomposition of the real and non-singular deformation mapping into orthogonal and symmetric components. Thus: ^^-+8, oXj
(5.15)
where: rkirkj = Sy and qtj = qn
(5.16)
In practice, the values of the rotational matrices are estimated during the previous step of the analysis, so the right-hand side of equation 5.13 is a linear function of displacement gradients. It can be seen from this equation that the LCR increment of strain is the same as the infinitesimal value when the material is not rotating. Furthermore, the tensor r,y - 8^ is approximately skew-symmetric providing the incremental angles of rotation of the material do not exceed about ten degrees. Thus for pure rotations up to this magnitude, equation 5.13 leads to approximately zero values of strain increment [5.2]. The importance of using the correct definition of strain increment is clearly demonstrated by figure 5.3. This shows two FE predictions (one with LCR strain, the other using the infinitesimal definition) of the force and stress during the combined extension and rotation of a single element considered earlier, and compares these with analytical values.
5.4 Elastic-plastic formulation
73
With the LCR definition of strain increment, and the deformation gradient defined in equation 5.11, equation 5.8 may be rewritten in its incremental form: 8(AdIm)AfIm=
(5.17)
5.4
ELASTIC-PLASTIC FORMULATION
5.4.1
Yield criterion
So far, no special assumptions have been made about the nature of the material being deformed, and hence about the form of the constitutive matrix Dijki. It is now necessary to obtain the particular constitutive matrix that may be applied Fig. 5.3 Comparison of analytical values of stress and force with FE predictions using infinitesimal and LCR definitions of strain increment: theoretical; O FE - infinitesimal-strain; • FE - LCR strain.
1
1
500 I)
0 -
*•
'
1
1
a
B
—. — o
1
j
1
—•
B—
o
-1000
°n y F < -1500
o -
r S
-2500
o O
-2000>
':
o ~
100
,6
0
o - —B-—
-
-500 •8
1
1
.
1
!
i
A
.
i
1
a
-
1
O
0
-
-100
-
-200
-
-300
-
-400
-
-500
-
H
.— o —
,
I 1 "
'
B—
, i
B—
o
, •
O
-
o
-
o
-
o
O
.
10
i
20
.
-
e—
o
-600 ! 0
, '
i
30
I
40 50 a (degrees)
1
60
,
1
70
,
i
80
«
90
74
Finite-strain formulation
to the deformation of metals. Clearly, the general theory may just as easily be applied to the deformation of other materials, such as rocks or soils, provided that the appropriate constitutive matrix is defined. Most common metals appear to obey the von Mises yield criterion described in Chapter 3. Previously, we have assumed that yield stress depends only upon the value of plastic strain. Now we consider the more general statement that a region will deform plastically when its generalised stress a reaches a critical value determined by the accumulated plastic strain e p , the strain rate e p and the temperature T, that is when: o2 = | o { / a { / = Y 2 ( ? , l M )
5.4.2
(5.18)
Elastic-plastic flow rule
As mentioned in Chapter 3, a basic assumption of a non-viscous elastic-plastic formulation is that an increment of strain may be divided into its elastic (recoverable) and plastic (irrecoverable) parts. This assumption appears to be valid providing the increments are not too large [5.3]. Normality of the plastic strain increment to the yield locus in stress/strain space [5.4], and the use of the generalised form of Hooke's law for the elastic component leads as before to the Prandtl-Reuss flow expression [5.5]: ((l
5.4.3
A
S
A
d
A
'
(5.19)
Elastic-plastic constitutive relationship
The Prandtl-Reuss equations may be rearranged [5.6] to obtain the incremental form of equation 5.7. This is described in Appendix 3. From equation A3.18 the elastic-plastic constitutive matrix Dijki is defined to be: Duki — 2G [ 8ik 8ji + 8ij 8ki I
I
In practice, the last term of equation 5.20 is omitted if an element is deforming elastically.
5.4.4
Effect of plastic incompressibility
In rigid-plastic and similar FE formulations, an explicit constraint must be imposed upon the solution in order to enforce volume constancy during plastic deformation. There are several methods of doing this, one example being the penalty-function technique [5.7]. All of these can be shown to be equivalent to multiplying the bulk-strain terms in the governing equations by a large number.
5.4 Elastic-plastic formulation
75
With an elastic-plastic formulation however, plastic incompressibility is an implicit assumption of the Prandtl-Reuss flow rule (equation 3.18). As the governing equations stand, this tends to enforce plastic volume constancy at every point of these elements. Since each element has only a limited number of degrees of freedom, the effect is to over-constrain the deformation [5.8]. In the actual workpiece, the condition of volume constancy will apply throughout the plastic region, thus introducing one constraint for every yielded particle, each of which has three independent components of displacement. This ideal ratio of 3:1 for the number of degrees of freedom to the number of volume constraints is rarely met in an FE model, unless special techniques are used. For example, in an FE mesh there are three degrees of freedom for each node. The ratio of degrees of freedom to volume constraints will therefore depend upon the number of constraints per element and upon the ratio of the number of nodes to the number of elements in the mesh. Both of these quantities depend upon the type and arrangement of the elements. Thus in a regular mesh of 3-D eight-node linear elements, the numbers of nodes and elements are approximately equal, provided that the number of elements is large, and there are seven volume constraints per element. The constraining ratio is therefore 3:7. A regular 3-D 20-node quadratic element has sixteen volume constraints and, when the number of elements is large, there are four times as many nodes as elements in a regular mesh. The constraining ratio in this case is therefore 3:4. In neither of these examples does the constraining ratio approach the ideal value. Since the nodal displacements in an FE mesh have to satisfy so many additional constraining conditions imposed by the requirements of volume constancy, the FE solution tends to predict unrealistic modes of flow, particularly when plasticity is fully developed. This requires some relaxation of the constraints.
5.4.5
Element-dilatation technique
The element-dilatation technique, first proposed by Nagtegaal, Parks and Rice [5.9], resolves the problem of over-constraint by separating the expression that determines the volume change out of the governing equations. This term may then be modified to produce the required number of volume constraints per element. From equation 5.20 it can be shown that: 8(Ae^DijklAekl
= ^ A c ^ D ^ / A e ^ + K'8(Aepp)Aeqq
(5.21)
in which the deviatoric strain increment is defined by: Ae'tj= Aeir
^Aep,
(5.22)
76
Finite-strain formulation
and K is the elastic Bulk Modulus E/3(l-2v) (Appendix 7). It can be seen that the last term of equation 5.21 represents the contribution of the bulk or volume strain to the element stiffness. The bulk-strain increment is defined to be an independent function Ac)), called the dilatation increment, of position: Aejj = A <#*,-)
(5.23)
The form of the dilatation function depends upon the number of volume constraints that are required per element. For example, a regular mesh of a large number of eight-node linear elements requires one extra equation to be satisfied per element to give the desired ratio of degrees of freedom to volume constraints. Thus the dilatation function must have only one coefficient and so be constant throughout the element. The sole volume constraint is then the condition: V = V-A<j>
or:
A^yjA^dF
(5.24)
(5.25)
Alternatively, an extensive regular mesh of 20-node quadratic elements requires four volume constraints per element, so the dilatation function in this case must contain four independent coefficients defining a tri-linear function of local co-ordinates Xt\ Acf>= A >o + AQiXt
(5.26)
(The local co-ordinates of a hexahedral element are defined to be plus or minus one at the corners of the element. The origin of this co-ordinate system is at the centroid of the element.) Multiplying equation 5.23 by each of the local co-ordinates (or by one) and integrating with respect to the volume V* of the element in the local co-ordinate system gives four constraining equations to be satisfied for each element: 8
(5.27)
Using equation 5.23, equation 5.17 may be re-written as: 8(AdIm)AfIm
=
Since most of the examples of 3-D FE analyses presented in this work use eight-node linear elements, it will henceforward be assumed that one volume constraint is to be introduced into the expressions governing the deformation of
5.5 Element expressions
77
each element. Most of the following theory would still apply if more than one volume constraint were required, though the resulting equations would be slightly more complex. Substituting equation 5.25 into equation 5.28 produces: 8(AdIm)AfIm
= J(s(Ae t j ){D i j k l - 28]l(rik)Aekl
+
8(Aujti)
)Aejj \ dV+ -^ J 8(Aeu)dV' J A^dV
5.5
ELEMENT EXPRESSIONS
5.5.1
Interpolation of nodal displacement
In order to evaluate the volume integrals in equation 5.29 it is necessary to express the strain increment and the deformation gradient at each point in terms of the nodal displacements. The incremental displacement of a particle of the element with initial co-ordinates Xj may be written in the form: Aut = N^Adn
(5.30)
where 7V7 is the interpolation function of the /th node of the element which was introduced in Chapter 2 (equation 2.10). If:
Nu= —i-^ll
(5.31)
oX[
then from equation 5.11: Auu
= NItjAdn
(5.32)
and from equation 5.13: Aeij = BijImAdIm=
—{B
ijlm
+ BjiIm)Adlm
(5.33)
where BijIm is the LCR strain increment/nodal displacement increment matrix and: Bijim = rmiNItj- -jVik^Nw-StJV/J
(5.34)
The exact nature of the interpolation functions depends upon the number and arrangement of the nodes of the element. Usually, they may be expressed most simply as functions of a local co-ordinate system defined for each element. In this case, the Cartesian spatial derivatives of the shape functions in equations 5.32 and 5.34 must be evaluated from the local spatial derivatives using the chain rule.
78
Finite-strain formulation
5.5.2
Incremental element-stiffness equations
Substitution of equations 5.32 and 5.33 into equation 5.29 produces: 8(AdIm)Aflm
=
8{AdIm)\\Bijlm(Dijkl-28jl(Tik)Bkljn
NI,i8mno-ikNJ,k
-K'BiilmBjjJn
J AdJndV
(5.35)
8{AdIm) ( -£ | BiiImdV-1 BjjJndV J AdJn from which the arbitrary variations in nodal incremental displacement may be cancelled to give the element stiffness equations: Afim = (K]mjn + K°lmJn + KJmJn)AdJn
(5.36)
K€imjn = \BjjImDijkiBktJndV
(5.37)
where:
is the element deformation stiffness matrix, i.e. the usual infinitesimal-strain matrix, KalmJn = j(Nu8mn
- 2BijIm8j,cTikBkljti)dV
(5.38)
is the element stress-increment correction matrix and:
1 jjJ
- 1 BiiImBjjJndV
)
(5.39)
is the element dilatation correction matrix. The volume integrals in equations 5.37-5.39 are most conveniently evaluated by Gaussian quadrature. To this end, the integrands may be multiplied by the Jacobian of the mapping of local to global co-ordinates to enable the integration to be carried out with respect to the volume of the element in local co-ordinates. As described in Chapter 2, the global stiffness matrix is obtained by assembling the stiffness matrices of all the elements in the mesh.
References [5.1]
Lee, E.H. The basis of an elastic-plastic code. SUDAM rep. no. 76-1, Stanford University (1976). [5.2] Pillinger, I., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. A new linearized expression for strain increment for the finite-element analysis of deformations involving finite rotation. Int. J. Mech. Sci. 28, 253-62 (1986). [5.3] Lee, E.H. and McMeeking, R.M. Concerning the elastic and plastic components of deformation. Int. J. Solids Struct. 16, 715-21 (1980). [5.4] Drucker, D.C. A more fundamental approach to plastic stress-strain re-
References
[5.5] [5.6] [5.7] [5.8] [5.9]
79
lations. Proc. 1st National Congress on Applied Mechanics, ed. E. Sternberg, ASME, pp. 487-91 (1951). Hill, R. The Mathematical Theory of Plasticity. Clarendon Press (1950). Yamada, Y., Yoshimura, N. and Sakurai, T. Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by the finite element method. Int. J. Mech, Sci. 10, 343-54 (1968). Price, J.W.H. and Alexander, J.M. Specimen geometries predicted by computer model of high deformation forging. Int. J. Mech. Sci. 21, 417-30 (1979). Nagtegaal, J.C. and de Jong, J.E. Some computational aspects of elasticplastic large strain analysis. Computational Methods in Nonlinear Mechanics, ed. J.T. Oden, North-Holland, pp. 303-39 (1980). Nagtegaal, J . C , Parks, D.M. and Rice, J.R. On numerically accurate finite element solutions in the fully plastic range. Comp. Meth. Appl. Mech. Eng. 4, 153-77 (1974).
6
Implementation of the finite-strain formulation
6.1
INTRODUCTION
The previous chapter was concerned with producing a correct statement of the elastic-plastic element stiffness equations and their assembly into the global stiffness equations. As they stand, these expressions simply describe how the forces applied to the nodes of a discretised model of the workpiece change as those nodes are displaced by small amounts from their starting positions. It is now necessary to consider how these equations can be used within an FE analysis for the study of practical metalforming operations.
6.2
PERFORMING AN FE ANALYSIS - AN OVERVIEW
It is instructive to consider the entire process of performing an FE analysis of a metalforming operation (figure 6.1). Of the four main parts of this process, the FE calculation will always be undertaken by a computer program: the description of the metalforming operation and the pre-processing and post-processing stages may be integrated into the FE package, may be separate computer programs, or even be performed by hand. The starting point in any such analysis is the metalforming operation itself. The first stage in the analysis is therefore concerned with obtaining a complete description of the operation in geometric or numerical form. This description will include information about the initial geometry of the workpiece, the shape of the dies and how the relative position and orientation of the dies and workpiece change during the deformation, the previous history of the workpiece and the dies, and the particular metal being formed. Although not often considered when discussing FE procedures, this particular
6.2 Performing an FE analysis - an overview
81
stage is just as much a part of the analysis process as the FE calculation itself, for the data used by the numerical calculation cannot be determined until the different aspects of the metalforming operation have been rigorously described. This preliminary stage is usually performed manually, but in principle the process could be automated to a certain extent by means of an expert system [6.1]. Fig. 6.1 Schematic representation of the FE analysis process. ETALFORMIN OPERATION
DESCRIPTION s^r-r-. "x , OF ( Workpiece \ ( Die OPERATION V Pre- form J V geometry
PREPROCESSING
Generate mesh
Determine initial boundary conditions
stermine change in boundary conditions with time
Input and check data
Calculate incremental displacement of external nodes
FE CALCULATION
Repeat for required no. of increments
Assemble and solve stiffness equations to obtain incremental displacements of all nodes and external reactions Calculate strain and stress increment and deforming loads Update mesh geometry, state parameters and boundary conditions Print or save current workpiece description
POSTPROCESSING
Interpret results
Determine initial' state parameters
Select from material database or fit function
82
6.3
Implementation of the finite-strain formulation
PRE-PROCESSING
The pre-processing procedures take the description of the metalforming operation and express it as numerical data that may be input into the FE program.
6.3.1
Mesh generation
An FE mesh is generated by dividing the volume of the workpiece into a number of elements, joined together at nodes. The mesh is then defined to the FE program by specifying the nodal co-ordinates and by listing the nodes belonging to each element, usually in some standard sequence that determines the topology of the element. As a general rule, and perhaps with the exception of the constant-strain triangle in 2-D and the tetrahedron in 3-D, the commonly-available elements are all suitable for performing metalforming analyses. Although the different elements may lead to different results [6.2], providing the correct formulation is used the differences will not be large. This is in contrast to elastic FE analyses in which it may be of vital importance to choose the correct type of element for a particular problem. That this should be so is perhaps not surprising: the accuracy of the metalforming analysis is mainly a result of obtaining a correct variational principle governing metal flow, valid for any discretisation, and not in choosing a particular set of element degrees of freedom and interpolation functions to model a given strain and stress distribution. Up to a certain point, the accuracy of the solution improves as more elements are included in the mesh, but in practice the constraints of available computer memory or time will put a limit to the number that can be used. It is desirable, therefore, to use a large number of smaller elements in the regions of the workpiece which are expected to undergo severe deformation. The generation of meshes with non-uniform element size is made easier if a selection of different element types is available within the FE program, though this is not essential. The mesh can be generated manually, and the numerical information obtained typed into the computer, but for any except the smallest meshes this is very time-consuming and prone to error. The alternatives are either to write a computer program to generate the mesh or to use a commercial mesh-generation package, if a suitable one is available. The latter option would certainly be preferable if it is intended to examine a large number of different workpiece geometries. The disadvantages of this approach are that it may take a long time to learn to use commercial mesh generators properly, that it is sometimes difficult or impossible to obtain a mesh with precisely the required variation of element type and resolution, and that these programs are expensive. Commercial programs may contain many thousands of lines of computer code, and yet by taking advantage of the characteristics of a particular geometric
6.3 Pre-processing
83
configuration, it is possible to write a program containing only a few hundred lines that can produce similar or better results. Such programs will generally be much easier to use and can of course be tailored to produce exactly the mesh required in any particular instance. If all the workpieces to be modelled are members of a small number of geometric classes, it will be possible to compile a suite of such specially-written programs.
6.3.2
Boundary conditions
Ttere are two methods of defining mechanical boundary conditions - by means of nodal constraining conditions and by modelling die surfaces. Nodal constraining conditions specify how nodes are to move (or not to move) throughout the deformation. These would be used, for example, to make a face of the FE mesh act as a plane of symmetry of the workpiece (figure 6.2). A constraining condition will determine which of the three Cartesian components of displacement are to be unconstrained (i.e. be obtained as solutions of the stiffness equations), and which are to have some specified (frequently zero) incremental value. Since nodes will not always be constrained upon planes that are aligned with the global axis system, it is necessary to be able to specify a nodal constraining condition with respect to a locally rotated set of axes (Appendix 6). In practice, it is often found that many nodes are subject to the same constraining condition. It is therefore convenient to define each constraining condition once only, and then to specify the constraint applied to a given node using the constraining-condition number. Nodal constraining conditions apply throughout the deformation, but the bounFig. 6.2 Modelling planes of symmetry using constraining conditions. unconstrained *** zero displacement
unconstrained zerodisplacement
84
Implementation of the finite-strain formulation
dary conditions resulting from contact between the workpiece and the dies will change as the metalforming operation proceeds. To determine the boundary conditions at any part of the outside of the FE mesh at any stage it is necessary to determine which nodes are in contact with the dies. The shape and position of the dies must therefore be made known to the FE program. The easiest way of doing this is to model each die by means of a set of primitive geometric surfaces [6.3]. In this way the determination of nodal contact is much simplified. All but the most complex die surfaces can be described using very few primitive shapes. Figure 6.3 shows examples of some of these - other simple geometric surfaces may also be found useful. The description of the die surface will usually contain information about the Fig. 6.3 Primitive geometric surfaces used to model dies.
+ve
6.3 Pre-processing
85
frictional restraint resulting from the particular lubricant used (if any). The heat transfer coefficient of this layer may also be specified. As an alternative to using geometric primitives, the dies could be modelled by means of additional finite elements. In addition to permitting contact between the workpiece and the dies to be determined, this would also allow the deformation of the dies to be investigated, but with added difficulties due to the complex nature of the contact between the two surfaces. This approach will not be considered further here.
6.3.3
Material properties
These include the elastic coefficients (Young's Modulus, Poisson's Ratio), the thermal parameters (conductivity, specific heat) and the yield stress. For the purposes of the FE calculation, the elastic properties may be taken to be constant. However, it may be necessary to specify the dependence of the thermal properties upon temperature, and the yield stress will certainly vary with strain, and probably with strain rate and temperature as well. The instantaneous values of these variables may either be calculated from some specified function, or be calculated by interpolation between empiricallyderived data. The former process is much simpler but it means the FE analysis is restricted to those materials for which suitable functions have been written into the computer program. The method of data interpolation is more flexible but may require a large amount of information to be specified, particularly if the yield function is to be considered as a function of three variables. The process of interpolation in this last case is not trivial, though it is a standard mathematical technique and library routines are available. Fortunately only the values of yield stress and its derivative with respect to strain are required, not the derivatives with respect to strain rate and temperature.
6.3.4
Deformation
One way of specifying the deformation for the FE calculation program has already been described, namely by prescribing values for the components of incremental displacement of particular external nodes (Section 6.3.2). In general, it is not known beforehand exactly how a particular node, even an external one, will move during the metalforming operation. What is known is how the dies move. So in addition to specifying the shape and initial position of the die surfaces, it is necessary to provide information to the FE program about how these positions change during the metalforming operation deformation. The simplest way is to specify the incremental displacement of each of the die
86
Implementation of the finite-strain formulation
surfaces. The specification of incremental rotations may also be necessary (figure 6.4) in order to model, for example, the rolling process. This information allows the FE program to determine two things: where each primitive boundary surface is at the start of each increment, and the nature of the constraints to be applied to any nodes in contact with these surfaces (figure 6.5). By specifying the displacements of boundary surfaces and boundary nodes with respect to an arbitrary increment of the deformation, it is possible, if a simple approach is required, to ignore the influence of time entirely. On the other hand, time can just as easily be included in the formulation by specifying a time step for each increment. This time step may be constant, or it may be calculated in accordance with the sinusoidal variation of deformation with time typical of a mechanical press. It isnalso possible to model more complicated press characteristics. Fig. 6.4 Incremental movement of boundary surface.
rotation
displacement
Fig. 6.5 Constraint of node in contact with moving boundary surface. P is position of node at the start of the increment; P' is the position of node at the end of the increment for sticking-friction conditions. plane of constraint of node
6.4 FE calculation
6.3.5
87
Initial state parameters
In order to evaluate the FE stiffness equations, and hence to calculate the pattern of nodal displacements, it is necessary to know the geometry of each element and the values of stress, strain, strain rate, temperature and incremental rotation throughout the FE mesh. The description of the initial mesh geometry has been considered earlier; the initial values of the other quantities must also be specified to the FE program. The temperature distribution resulting from the pre-heating of the workpiece, or the stress and strain distribution arising from a prior stage in the forming sequence may both be modelled quite easily. Initial values of the state variables may also be calculated as a result of a re-meshing procedure (Section 6.6.1).
6.4
FE CALCULATION
The main parts of the FE program are shown in figure 6.1. These parts will now be considered individually in more detail.
6.4.1
Input of data
The pre-processing stage of the FE analysis will have produced a file of numerical information precisely defining the deformation process. This information must now be read into the FE program. It is sensible at this stage to check the information for error. However, the extent of the checking is open to interpretation.Thus, the internal consistency of the data, such as whether an element has the correct number of nodes, or whether all the nodes have been defined, is easy to verify, and it is fairly straightforward to pin-point values which could lead to machine overflow or other errors of calculation. It is not so easy to determine whether the information supplied is 'sensible'. Thus an FE program designed to look at forging in the automobile industry could quite reasonably consider that any co-ordinate measured in metres rather than millimetres must be a mistake. This will cause problems if the program is later used by someone to model the forming of a marine propellor shaft. Clearly, the program could be easily modified, but it illustrates the difficulty of trying to impose sensible limits on the values of the data without unduly restricting the application of the program. Providing such unusual values do not jeopardise the FE calculation, it is probably best just to issue warnings that the data is suspect and carry on regardless. It goes without saying that such warnings, as is the case with all error messages produced by the FE program, should endeavour to describe fully not just what is wrong, but what exactly has caused it and, if possible, what can be done to put it right.
Implementation of the finite-strain formulation
However much error checking takes place, it is customary to echo the data as soon as they are read in to provide a record of how the program has interpreted the data file.
6.4.2
Displacement of surface nodes
The FE analysis is carried out incrementally, and at the start of each increment it is necessary to determine the values of any prescribed components of incremental nodal displacement. A component of displacement of a node may be prescribed either because a constraining condition has been specified for that node, or because the node is in contact with a die surface, or for both these reasons. At this stage in the FE calculation therefore, the program must check the position of each external node with respect to each of the specified primitive boundary surfaces. Any node which appears to have passed through a boundary surface must be re-positioned on the surface (figure 6.6). If the die surface is stationary, the node should be constrained to move within a plane that is tangential to the surface at point P (the point of contact of the node with the surface at the start of the increment - figure 6.5); if the die surface is moving, the node is constrained to move within a plane passing through the point P' that P will occupy at the end of the increment. The normal n of this plane of constraint is the average of n and n', where n is the normal to the surface at P at the start of the increment, and n' is the normal to the surface at P' at the end of the increment. This rather complicated method of constraint is necessary in order to be able to deal with rotation as well as displacement of boundary surfaces. To take the two extremes, it can readily be seen that the plane of constraint is a tangent to the displaced boundary surface if it does not rotate, and that if a cylindrical surface is subject to a simple rotation about its axis, the plane of constraint is a chord to the surface passing through P and P' (figure 6.7). The situation just described, where the node is constrained to move within a plane, will apply whenever a free-surface node comes into contact with just one Fig. 6.6 Checking for contact between FE mesh and a boundary surface: (a) at the end of an increment, (b) after checking for contact with dies. - die surface
(a)
(b)
6.4 FE calculation
89
primitive boundary surface. If the node was originally subject to a constraining condition (figure 6.8) or if the node is found to be in contact with more than one boundary surface (figure 6.9), then the constraint applied to the node will need to satisfy, if it is possible, all the conditions imposed upon it. This may result in the node being made to move along a line* of intersection or, in the extreme case, being fixed in position upon the die surfaces. In general though, a node in contact with a die is allowed to move freely parallel to the die surface throughout the increment. No restriction is otherwise placed upon this movement; any frictional restraint applying to the die surface is imposed by a separate technique (Section 6.4.3.1). One final point should be noted in connection with the boundary-surface technique. This is that once a node has made contact with a die surface, the method outlined above will tend to keep it in contact throughout the rest of the deformation. The method should therefore also check whether the stress acting Fig. 6.7 Constraint of a node in contact with (a) displaced die, (b) rotating die. (a)
(b)
Fig .6.8 Contact of die with node subj ect to previous constraining condition. V/
^\j^
boundary
V
unconstrained zero displacement
incremental displacement
90
Implementation of the finite-strain formulation
normally to the surface of the mesh at a given node is tensile, indicating that there is a tendency for the node to pull away from the die. If this is the case, the node should not be constrained upon the die surface during the next increment, even if it appears to be in contact with it.
6.4.3
Assembly and solution of the stiffness equations
The basic principles of the assembly and solution of the stiffness equations have been discussed in Chapter 2. However, several other aspects of the solution technique need to be considered in a practical metalforming program. 6.4.3.1
Incorporation of frictional restraint
The frictional restraint acting at the interface between the workpiece and the dies is an important factor in determining the pattern of flow, strain and stress in the component. In metalforming operations, the condition of macroscopicallyrigid sliding surfaces is no longer met, and the familiar proportionality between the frictional force opposing the relative movement and the normal reaction no longer applies. Instead, for a given die/workpiece interface and lubricating condition, it is found that, over most of the region of contact, the shear stress at the interface is an essentially-constant fraction m of the shear yield stress of the metal undergoing deformation. In practice this is an approximation, since the lubricant is Fig. 6.9 Constraint of nodes in contact with arbitrary planes p1, p2 and p3.
zero displacement — unconstrained zero displacement node on p 2 only
node on intersection ofp 1 ,? 3
6.4 FE calculation
91
affected by the interface pressure, but the assumption is good enough for most purposes. Thus, at one extreme, if the friction factor m equals zero, there is no tagential force acting at the interface and the sliding of the workpiece against the die is unimpeded. On the other hand, if m equals one, the plastic deformation of the workpiece at the interface must be due solely to a process of shear parallel to the die surface. Under these conditions, there can be no relative movement between the die and the material actually in contact with it. As might be expected, these two idealised conditions of zero and sticking friction are not met in practice and the friction factor m is found to approach, but not equal, the two limiting values. The actual value is commonly determined from laboratory trials using the ring test [6.4]. If the direction of sliding of the surfaces of the workpiece with respect to the die were known from the start, the imposition of a frictional traction force would be a simple matter. In a few simple geometries the direction is indeed known, and in others experiments may be carried out to determine this information, but this is not possible if the FE program is to be generally applicable and fully predictive, for example in the ring test itself. A method of applying a frictional restraint without prior knowledge of the pattern of flow is provided by the friction layer technique [6.5]. This requires a fictitious layer of elements to be created at the interfaces between the workpiece and the dies (figure 6.10a). The boundary technique ensures that the interface nodes can only move tangentially to the die surface, and the extra friction-layer nodes are prevented from moving in this direction. (If the dies are moving, of course, both sets of nodes may have the same superimposed motion.) The stiffness matrix of each friction-layer element is then multiplied by a factor, the Stiffness-Matrix Multiplier (SMM), that is proportional to m/(l—m) (figure 6.10b). Since the friction-layer nodes cannot move parallel to the die surface, the effect is to apply a shear force to the interface nodes acting in the opposite direction to their movement. As m tends to zero, the SMM, and hence this shear force, also tends to zero. As m tends to one, the SMM tends to infinity and the friction-layer elements become very stiff, essentially preventing any tangential movement of the interface nodes and thus sticking them to the die. The friction layer is fictitious. It is not actually modelling a lubricant but is merely a mechanism for modelling the effects of such a lubricant. Its existence is also transitory. Each friction-layer element can be created when the stiffness matrix of the associated interface element is evaluated. The stiffness matrices of the interface element and the friction-layer element may be assembled together and a Gaussian reduction used to eliminate the friction-layer nodes from these equations (for the displacements of these nodes are known) before the equations
92
Implementation of the finite-strain formulation
of the interface element are assembled into the global matrix. Thus the frictionlayer nodes need never form part of the main FE calculation. 6.4.3.2
Solution techniques
For reasons of economy, it is important to be able to use large step sizes in the FE analysis of metalforming operations. Many of the examples in this book have used nominal deformation increments of 1 or 2%. Since the FE stiffness equations are non-linear, the determination of the nodal displacement increments requires the use of some sort of iterative solution procedure. For example, in the initial-stiffness technique [6.6], which is illustrated in figure 6.11, the set of incremental nodal displacements (represented in the diagram by A d(1)) are obtained as usual by solving the global stiffness equations for the set of applied incremental forces (A/ 1} ) incorporating into the solution any prescribed values of nodal displacement. The increments of strain, and hence stress (Acr^) throughout the mesh may then be calculated, and the incremental form of equation 5.4:
$(dIm)fIm = l8(uhl)SljdV
(6.1) 1}
used to determine the values of incremental nodal force (A/ *) that are in equilibrium with the calculated nodal displacements. Fig. 6.10 Friction-layer technique. die surface
friction-layer nodes
frictionlayer elements billet interface elements
s
v
secondary surface die surface stiffness approximately zero
(b)
6.4 FE calculation
93
Because of the non-linearity of the stiffness equations, these nodal equilibrium forces will not, in general, be equal to the actual applied values, but the difference between them, the out-of-balance or residual forces (A/ 2 ) ), may be used to calculate a set of corrections to nodal displacement (AS 2) ) by solving the original stiffness equations again, this time setting any prescribed components of nodal displacement to zero. The increment of nodal force (A/ 2) *) in equilibrium with the displacement correction can then be found, and compared with the first set of residual forces (Af2^). The difference may then be used to calculate a second set of corrections to the nodal displacements (Ad^), and so on. This process may be repeated until the values of the residual forces or displacement corrections are smaller than some required limit. The advantage of this technique is that the global stiffness matrix needs to be calculated and inverted only once, at the start of each increment, so each subFig. 6.11 Schematic representation of initial-stiffness iteration. / Af=KAd ill
Ad™
94
Implementation of the finite-strain formulation
sequent iteration may be performed in a fraction of the time required for a full solution of the stiffness equations. The disadvantage is that many iterations may be required for the convergence of the solution. Occasionally, the solution may not converge at all. In an alternative form of iteration, the global stiffness matrix is re-calculated at the end of each iteration; although this means more computation at each step, convergence is usually quicker. A good example of this approach is provided by the secant-modulus method of solution, illustrated in figure 6.12. This is a second-order Runge-Kutta technique in which the stiffness matrix is evaluated using the material stress and strain at the start of the increment (represented by a° and e° in the diagram). Fig. 6.12 Schematic representation of secant-modulus technique. Af=K(o°,e°)Ad
6.4 FE calculation
95
The equations are then solved for the set of applied forces A/to obtain a first approximation to the set of incremental nodal displacements (Ad a ). These may be used to estimate the stress and strain half-way through the step (o^e" 1). A new stiffness matrix is then constructed, based upon these mid-increment values, and solved to give a better estimate of the incremental nodal displacements (Adc) and the strain (ec) and stress (cf) at the end of the increment. 6.4.3.3
Yield-transition increments
A more extreme form of non-linearity occurs when an element of the workpiece yields and starts to deform plastically, for at such times the elastic constitutive matrix in equation 5.37 needs to be replaced by the elastic-plastic one, with a correspondingly-large change in the magnitude of the deformation modulus. In order to overcome this problem, it is possible to reduce the size of yieldtransition increments automatically [6.7], so that elements always yield at, or immediately before, the end of a step.
6.4.4
Updating of workpiece parameters
As shown in Chapter 5, the FE stiffness equations depend upon the current geometry of the mesh and the current distributions of stress, strain, strain rate, temperature and element rotational values. These quantities must therefore be updated at the end of each increment. In addition, since the boundary conditions for the solution of the equations change throughout the deformation, these must also be re-determined before the next increment of the analysis, and the values of force acting upon the surface of the workpiece need to be calculated in order to be able to assess die stresses and forming loads. Updating the mesh geometry is simply a matter of adding the incremental nodal displacement values to the nodal co-ordinates. Updating the other quantities will be considered in more detail. 6.4.4.1
Strain components and rotational values
The incremental-strain tensor may be evaluated at any point within an element using the expression: (6.2) A / /
2
rik><
Au
i>k>
)
which was introduced in Chapter 5. The deformation gradient Aw/;7 may itself be obtained from the known values of AdIh the incremental nodal displacements for that element, using: AM/,/ = NhjAdn
(6.3)
As explained in Chapter 5, the rotational matrix rtj is defined by the expression: rikqkj = Auitj
+ 8ij
^'^
96
Implementation of the finite-strain formulation
Because rtj is an orthogonal matrix (equation 5.16), multiplying each side of this expression by its transpose gives: nmqmirikqkj
= 8mkqmlqkj
= (AM,,/ + dn)(Auiyj + $,-)
(6.5)
Denote the right-hand side of equation 6.5 by the matrix htj. Then since qtj is symmetric: qikqkj = hij
(6.6)
htj may be evaluated quite simply from the known increments of nodal displacement Adn using equation 6.3, and equation 6.6 solved iteratively using Newton's method, to obtain qkj\
dtf = l ^ r ' + ^ r 1 )
(6.7a)
pir 1} ^r 1} = h
(6-7b)
where: and:
qif
=
wkj
(6.7c)
(l
in which q k) denotes the /th approximation to qkj. In practice, this process of iteration converges rapidly since the components of A utj are much smaller than one. The rotational matrix rtj may then be evaluated for use in equation 6.2, and subsequently in the stiffness formulation for the next increment, using equation 6.4. 6.4.4.2
Deviatoric stress
The elastic-plastic constitutive matrix (equation 5.20) should only really be used to evaluate increments of stress from increments of strain if the increment size is very small. This is because the Prandtl-Reuss flow expression (equation 5.19) on which this matrix is based, is strictly only defined in terms of infinitesimal deformation steps (equation 3.20a). If finite-sized steps are used, Nagtegaal and de Jong [6.8] have shown that the FE solution obtained may be inaccurate or unstable. Instead, the mean-normal method may be used to calculate the increments of deviatoric stress. This technique was first suggested by Rice andTracey [6.9] and extended to include strain-hardening properties byTracey [6.10]. The technique is illustrated in figure 6.13 with reference to stresses plotted in the synoptic plane. Points represents the stress state at the start of the increment on the initial yield locus, and B represents the final stress state after some degree of strain hardening. The form of the Prandtl-Reuss flow expression given in equation 5.19: Aeij = —{(l + v)Aaij-vdijA(Tkk)+A\crfij
(6.8)
6.4 FE calculation
97
assumes that the increment of plastic strain is parallel to the deviatoric stress at the start of the increment. The mean-normal method makes the more realistic assumption that the plastic strain increment is parallel to the deviatoric stress half-way through the step. It can be seen from figure 6.13 that the latter stress (OC) is parallel to cr'ij (OD) where: I (6.9) and from the generalised form of Hooke's law: ?/ =2G(Aeij+8ij
(6.10)
(-^
The modified form of the Prandtl-Reuss equations given in equation 6.8 is therefore: A€iy = — ((l + i/) Aay-vSijAauc) + - ^ ^ (6.11) which may be rearranged to give: Ao-ij = Aofj - Amcr'ij
(6.12)
Equation 6.12 allows the stress increment to be calculated once the proportionality factor Am is known. Application of von Mises's yield criterion to the state of stress at the end of the increment shows that this factor may be evaluated by solving the equation: —3G
Fig. 6.13 Mean-normal method of calculating finite changes in stress.
Implementation of the finite-strain formulation
98
where: P
q=
(6.14)
=-o-ijO-ij
(6.15)
-
(6.16) The solution is carried out iteratively (figure 6.14), and is found to be rapidly convergent. The procedure is particularly efficient since the coefficients of the quadratic function need only be evaluated once at the start of the iteration. It can be seen that increment of generalised plastic strain is evaluated automatically as part of the mean-normal method. The increment of stress obtained by this method depends only upon the change in strain in the element. However, rotation of a stressed element will also cause the values of the stress components to change, even if there is no change in strain. Therefore, when updating the Cartesian components of total (Cauchy) stress, it is necessary to include any additional stress increment resulting from the element rotation. The importance of using an accurate method of calculating stress is illustrated in figure 6.15. This shows two FE predictions of the compression of a cube of Fig. 6.14 Iterative solution of mean-normal equations. /(Am) /(Am) = p(Am)2 + qAm + r
/(Am) = Y2( Am
\ Am<3> Am<2>
99
6.4 FE calculation
aluminium with zero inter-facial friction. The deformation ought, therefore, to be homogeneous. Figure 6.15a shows the unstable deformation resulting when the stress is calculated by a simple tangent-Z)-matrix approach; figure 6.15b shows the results obtained when the mean-normal method is used. 6.4.4.3
Hydrostatic stress
Changes in hydrostatic stress may be calculated by multiplying the changes in bulk strain by the bulk modulus. However, it is useful to have an independent means of evaluating the hydrostatic component of stress. Providing there are external surfaces of the FE mesh that are not in contact with dies, hydrostatic stress may be calculated from equilibrium principles. This is the indirect method proposed by Alexander and Price [6.11]. Since the present technique assumes zero body forces, principles of force equilibrium lead to the expression: - 3 ^
(6.17)
The deviatoric stress may be calculated at arbitrary points within the mesh using the mean-normal method, so the distribution, and hence the spatial gradients, of deviatoric stress may, in principle, be evaluated throughout the body. Integration of equation 6.17 along a given line therefore allows the change in hydrostatic stress to be calculated between its two end-points. Since the hydrostatic stress at any free surface is known to be equal to the negative of the deviatoric stress normal to that surface, a single starting point, Fig. 6.15 Zero-friction compression of cube: comparison of (a) tangent-Z)matrix and (b) mean-normal methods of calculating stress.
100
Implementation of the finite-strain formulation
or possibly several, can usually be found for the integration, so that the hydrostatic stress can be determined anywhere within the workpiece. As is often the case with the calculation of the gradients of a field variable, the calculation of the spatial derivatives of deviatoric stress is very susceptible to small errors in these stress components. If this is found to cause unacceptable errors in the hydrostatic stress distribution, then the estimates of the stress derivatives can be improved by performing a smoothing operation of some sort on the original stress values [6.12]. 6.4.4.4
External forces
Minimising the instantaneous rate of change of potential energy for an element (cf. equation 5.3 which is based upon the average rate in an infinitesimal step) leads to the following expression: lJ)cT
i j
dV
(6.18)
It should be noted that it is perfectly correct to use the Cauchy stress in this context because we are not considering any change in the stress state. Since Cauchy stress is symmetric, this may be written as: Kdim)fim ^{e^jdV
(6.19)
Using the interpolation functions Nh this gives the relationship: Kdlm)flm = Kdlm) j y
(S^NJJ
+ Sj^^jdV
(6.20)
or: (6.21) Equation 6.21 allows the forces acting upon any node of an element to be calculated from the previously-calculated values of element stress. If a node belongs to more than one element, the actual force applied to that node will be the sum of the values obtained for all these elements. Usually only the forces acting at the surface nodes are of interest: the forces acting at internal nodes should, because of equilibrium, all be zero. The integrals in equation 6.21, like the element-stiffness integrals, may be evaluated by Gaussian quadrature. 6.4.4.5
Strain rate
Since the yield stress may, in general, be a function of plastic strain rate, this needs to be calculated throughout the analysis. The plastic strain rate is simply the current incremental plastic strain divided by the time interval for the current increment. The disadvantage with this simple approach is that the value of strain rate is
6.4 FE calculation
101
only available after the calculation of deviatoric stress (Section 6.4.4.2), and this calculation itself requires that the current strain rate be known. An alternative then is to assume that the rate of plastic strain is the same as the rate of total (that is, elastic plus plastic) strain. In practice, there will be little difference between them, and it is a great advantage to be able to calculate the strain rate as soon as the increments of nodal displacement are known. In addition to simplifying the mean-normal calculation, it also avoids the discontinuous change in strain rate which would otherwise occur when an element begins to yield. The strain rate can most conveniently be calculated as part of the procedure which checks for transition between the elastic and plastic states and scales the increment size accordingly (Section 6.4.3.3) since this takes place immediately after the solution of the stiffness equations and must calculate the strain increment for its own purposes. As pointed out in Section 6.3.4, the actual time step associated with each increment of deformation may change throughout the analysis. 6.4.4.6
Temperature
The calculation of the temperature changes in the workpiece involves, among other things, the determination of the heat flowing throughout the mesh. Since this will itself depend upon the temperature distribution, the thermal calculation carried out during each increment of the deformation may need to be further divided into a number of steps. Let this number be N. Later, we shall examine a way of determining how many thermal steps are required. The change in temperature 8T of a given element during a thermal step will depend upon the change in energy of that element during the step: (Q cV\
QQ
) )
(6.22)
where c is the thermal capacity per unit volume of the material, Vis the volume of the element, 8Qd is the increase in energy of the element due to the work of deformation, 8Qf is the increase in the energy due to frictional work at external boundary faces, and 8QC is the change in energy due to the conduction of heat between the element under consideration and adjacent elements or dies. The last quantity may, of course, be positive or negative. Consider each of these contributions in turn. Work of deformation
The work of deformation per unit volume is the product of the average element generalised stress o-for the whole deformation increment, and the corresponding change in element plastic strain Ae p-
102
Implementation of the finite-strain formulation
Thus: 8Qd=—V-o-AeP
(6.23)
in which a is the proportion of the work of deformation converted into heat (an empirical quantity, usually in the range 0.85-0.95) [6.13]. Friction The frictional-work contribution depends upon the tangential force acting at any faces of the element that are in contact with die surfaces, and the relative movement of these surfaces. Suppose that F faces of the element are in contact with dies and let the index a refer to one of these faces. Since the shear stress at the boundaries is equal to the shear yield stress multiplied by the friction factor (Section 6.4.3.1) and, by the von Mises yield condition, the shear yield stress equals the yield stress in tension divided by the square root of three, then by assuming that half of the heat generated flows into the element, the frictionalwork contribution may be written as: 80'=— V N ^
maA YAda
° 2V3
(6.24)
where raa is the friction factor associated with face a of the element, Aa is its area, Ad a is the displacement of this face relative to the die surface during the whole of the deformation increment, and Y is the average yield stress of the element during the increment, a function of plastic strain, strain rate and temperature. Conduction of heat The contribution to the increase in energy of the element due to the conduction of heat through its faces is proportional to the duration of the thermal step At/N where Af is the time interval of the whole deformation increment. The temperature T at any point will be a function of its co-ordinates. Let x® be the global reference co-ordinates of the centroid of the element. If we assume the temperature at this centroid is the average element temperature, then applying the standard heat-flow expression to the element gives: . . , d 2T
At
where k is the thermal conductivity of the material. In order to evaluate the right-hand side of this expression, we shall assume that 3-D hexahedral elements are being used. The treatment for other types of element will be broadly similar. Define a local curvilinear axis system X1, with origin at the centroid of the
6.4 FE calculation
103
element, and with axes that pass through the centroids of the elements adjacent to the six faces of the element under consideration. Let the non-zero local co-ordinate of each of these adjacent centroids be plus or minus one (figure 6.16). If a particular face of the element under consideration is on the surface of the mesh, there is no adjacent element, so in this case let the axis pass through the centre of the face and have co-ordinate ±1 at this point. The six neighbouring points that, together with the centroid of the element under consideration, define the curvilinear axes, are labelled with a number between one and three and a plus or a minus sign, according to the ray of the local axis on which they are situated. Thus the reference co-ordinates of the neighbouring point (centroid or centre of face) with local co-ordinates (0,-1,0) will be x2-~, and so on. Since each of the three local axes is defined by three points (the centroid of the element under consideration and two neighbouring points) it will, in general, be parabolic in shape, and the value of any state variable may be approximated by a quadratic function of each of the three local co-ordinates. Thus the co-ordinates xt of any point within the element are given approximately by the expression: (6.26) where a^, btj are constants determined by the geometry of the local configuration. Similarly, the temperature Tmay be approximated by:
-Sj(Xjf
(6.27)
where T° is the temperature at the centroid. Fig. 6.16 Local curvilinear axis system for calculation of heat flow into an element. (0,0, l)T3+
(0, 1,0) T 2 +
d,o,o)r 1+
104
Implementation of the finite-strain formulation
The coefficients in equations 6.26 and 6.27 can be determined by evaluating these functions at each of the six neighbouring points: "ij=\
(x\+-x\-)
(6.28a)
bij = x\+ + jtj" - 2x°i
(V+-V~)
n=±
Sj
(6.28b)
(6.29a)
= 7J+_rJ-_2r°
(6.29b)
Now by the chain rule: d2T
s i n c e d2T/dJPdXk
=
d2T
,dXJ,2
d2Xj
dT
dXk
Oiij
Differentiating equation 6.26 with respect to the local co-ordinates produces: ^j=aij+biiX'
(6.31)
and by differentiating equation 6.27 we have: —== dX>
and
r.: + SjX> ' '
(6.32a)
- ^ 5 = s>
(6.32b)
Combining equations 6.29, 6.30 and 6.32, and evaluating at the centroid of the element under consideration: ^
) dxt I
)
2
k
dXidXk dx(
(6.33) where: C
=s
;it
+
. / BX* \ O~ =8ik 3
\ dxt )
2
r l d2xj
dx T
2 dXidXk dxt
atx?
(6.34a)
0
„ ^AU, (6.34b)
atjc?
The matrices of first-order partial derivatives in equations 6.34 are obtained by inverting the matrix of partial derivatives in equation 6.31; the second-order partial derivatives are obtained by differentiating these first-order derivatives
6.4
FE calculation
105
with respect to the local co-ordinates. When evaluated at the centroid of the element, which has local co-ordinates (0,0,0), these partial derivatives are found to be functions of just the quanitities atj and btj defined by equations 6.28. The coefficients O+ and O~ therefore depend only on the local geometry and so only need to be evaluated once for each increment of deformation, however many steps there may be in the thermal calculation. Returning to the heat-flow equation, the increase in energy of the element due to conduction of heat in one thermal step is:
$QC = ^YA1 ( Si+O+(V+-T°) + Si-O-(V--T°)
\
(6.35)
The temperatures in this expression refer to the start of the particular thermal step, and are updated at the end of each such step. The coefficients S*+ and S]~ in equation 6.35 allow the expression to be used even if the element has faces that are on the outside of the mesh. If neighbouring point/+ or/— is an adjacent element centroid: S'i+ = 1 o r S']~ = 1
(6.36a)
if neighbouring point /+ or /— is the centre of a face in contact with a die: I
K
}
_
+ i+
P k +k +
=_L_K
] ]] l]l~k ~k-+k -+k
V
'
]
where k* , k ~ are the heat transfer coefficients of the die/billet interfaces, and P+, P~ are the distances from the centre of face j+ or /— to the centroid of the element. Finally, if neighbouring point /+ or / - is a free-surface face: S'i+ = 0 or
Sj- = 0
(6.36c)
The expressions in equation 6.36b allow for the fact that the centre of the element face will not be at the same temperature as the die (V + or V~). Equation 6.36c ensures that there is no heat flowing through free-surface faces. The method just described can easily be modified, if required, to include the heat flowing due to convection at free surfaces. Similarly, it is possible to adapt the method to calculate the flow of heat through the dies themselves [6.14]. During each of the TV thermal steps of the deformation increment, the temperature change in each element is calculated using equation 6.22 and then the distribution of temperature throughout the mesh is updated for the next thermal step. The heat flowing at any instant between two isolated points of a medium is proportional to the difference in temperature of those points, and acts in such a way as to reduce this temperature difference. In the absence of any other effects, the temperature difference will tend to zero with time. Since the temperature difference becomes smaller all the time, a step-wise calculation will
106
Implementation of the finite-strain formulation
over-estimate the heat flowing between the two points. If these steps are too large, then not only will the corresponding estimates of temperature be inaccurate, but there is a possibility that the over-estimates of the heat flowing will actually lead to predictions of the temperature difference that diverge and change in sign between successive steps. For similar reasons, this type of instability, though of a much more complicated nature, can also occur in the FE thermal calculation if the size of the thermal step is too large. The critical step size is very difficult to assess accurately, but by making some broad simplifying assumptions, it is quite easy to estimate a lower bound to this quantity. Fortunately it does not matter in an FE program that this value is an under-estimate since the computational effort expended upon the thermal calculation is small compared with that required for the construction and solution of the FE stiffness equations. It is of no great consequence, therefore, if more thermal steps are used than are strictly necessary. The problem then is, given a time interval of At for an increment, to estimate the number of thermal steps, N, that this must be divided into in order to ensure firstly that the calculated temperature changes converge to finite values, and secondly that the errors in these values are smaller than some specified limit. Equation 6.22 may be re-written in the form: k c '
. . .
\ A^ ' / N
in which: | SQd + 8Qf | (6.38) cVAt \ ) For simplicity, assume that the temperatures of the neighbouring elements and dies are constant throughout the time interval At. If: k (6.39) Z = - (S^+O+V+ + Si~C*~Ti~) + P c and: P=
X = -(Sl+O+ + 5 j "C j ") (6.40) c then the estimated change in the temperature of the element during the ith thermal step is: (0 _ r o ( ;-i)
/z _
I
)
N
or: r o (O = z
^
+
(1
_ *Af } ro(/-i)
(6 42)
6.4
FE calculation
107
hence, the estimate at the end of N thermal steps is:
y Z X
N
X
and the estimated change in the temperature AT of this element is given by: Ar 0est
=
I
( Z
_ ^0(0)) [ 1 - ( 1 - ^ f
1
(6.44)
The correct value of the change in temperature in this simplified model is found by integrating equation 6.37 with the substitutions in equations 6.39 and 6.40: ( Z * T ° <0 ) ) ( l ~*A') (6.45) From equation 6.44, it can be seen that the estimated temperature difference will converge if: l_±£*f > - i (6.46) (Note, the left-hand side will always be less than +1.) Since various simplifying assumptions have been made, the number of steps required for a convergent solution is taken to be an order of magnitude larger than that defined by equation 6.46, i.e. it is required that: N > lOXAt
(6.47)
The absolute error in the estimated temperature difference may be found by comparing equations 6.44 and 6.45. If r is the specified maximum permitted error in the temperature difference, then it is additionally required that:
\z-xrm\r [ X
jTAyi N' J
A value of N is therefore chosen which satisfies equations 6.47 and 6.48 in every element of the FE mesh.
6.4.5
Output of results
The FE analysis is capable of producing information about many different aspects of the deformation. During each increment, the FE program calculates values of fundamental parameters such as nodal co-ordinates, and the distributions of stress and strain components, temperature and strain rate, as well as derived quantities such as nodal velocity, generalised stress and strain, hydrostatic stress, principal stress components, work rate and external loads. Whether the results are displayed numerically or in graphical form, the amount of information produced can be vast. If only a limited portion of this information is required, or if the results are only required at a few stages during the deformation, it is certainly possible to
108
Implementation of the finite-strain formulation
include numerical and graphical routines in the FE program. Generally, however, it is better to save a complete description of the current state of the FE model, in compact machine-readable form, during every increment of the analysis, and to use a post-processing program to display the results as required.
6.5
POST-PROCESSING
Post-processing simply means extracting from the large amounts of information produced by an FE program just those quantities that are of interest and either displaying these in some form or using them in an additional non-FE calculation. In practice, the information will be stored in a computer file, and so the postprocessing will be performed by a computer program, usually interactively. There are several advantages in using a post-processing program. Firstly, as mentioned previously, the output of the FE analysis becomes more manageable because only the particular information required need appear on paper. There is also the flexibility of being able to display the information in a variety of numericaland graphical forms - tables, charts, graphs, contour plots etc. Secondly, since the FE program will have saved results throughout the analysis, it is possible to examine in detail the deformation at any increment. Indeed, by interpolating between the results obtained for successive increments, it is possible to study any stage of the deformation. Thirdly, it is not necessary to know beforehand which quantities will be of interest in a particular process. Fourthly, since the post-processing information can be saved indefinitely, different aspects of the metalforming process can be studied at a later date, perhaps in the light of subsequent experimental findings. Lastly, the post-processing information can be used as input to other computer programs, such as CAD and control packages, and numerical-application programs. One such application is to use the FE stress and strain components throughout the deformation to predict when and where ductile fracture is likely to occur during the forging [6.15]. There are many commercial packages available for the analysis of information produced by FE prgrams. Most of these will have been designed for use in conjunction with elastic FE analyses, but some may be sufficiently flexible to be able to display the information relevant in the study of metalforming. Alternatively, it is not difficult to write very simple programs that can extract and output FE information in the required form.
6.6
SPECIAL TECHNIQUES
6.6.1
Processes involving severe deformation
Metalforming operations can involve total natural strains of unity or more. Depending upon the exact nature of the deformation, this may mean that, by
6.6 Special techniques
109
deformed grid re-meshed grid , _ .
_. 77%
T777
_ _
_
'!
.
1
--
-
.L. •
-
L- y-
r
•
;
!
,
r -
-
-
•
-
J-
-
J
\
j
-
-
,';-
backward extrusion
"
i -
r ~
L_
-~
!
\ -
(i-
.
;._ u . ;
-
\r. L _
• -
-
i_
•r
>
•
-
•
FE meshes
Fig. 6.17 Re-meshing of distorted FE grid during the analysis of backward extrusion.
the end of the FE analysis, the original mesh becomes so distorted that the interpolation polynomials are incapable of modelling the geometry of the elements and the associated state variables. If this is the case, a regular FE mesh must be re-defined within the current boundaries of the workpiece at intervals throughout the analysis (figure 6.17); on each such occasion the values of stress, strain, strain rate, temperature and incremental angles of rotation need to be transferred from the old to the new model using the element interpolation functions (figure 6.18). Fig. 6.18 Interpolation of nodal values during re-meshing.
\\w\\\ Fig. 6.19 FE analysis of steady-state rolling: region for which matrices solved (shaded).
6.6 Special techniques
111
For simple geometries, such as 2-D axi-symmetric or plane-strain deformations, it may be possible to perform the re-meshing automatically; more complex examples may require the FE mesh to be re-created manually, with perhaps the interpolation of variables being performed by a specially written computer program. When the FE mesh is re-defined in this way, the pattern of deformation is made much clearer if the geometry of the original mesh is updated along with that of the current FE mesh at the end of each increment - the super-grid technique [6.16]. This gives the impression that the original mesh has been used throughout the analysis and so more readily accords with experience gained from experimental observation. Some examples of this technique will be given later.
6.6.2
Steady-state processes
In order to use an incremental FE program to study steady-state forming processes, such as rolling, extrusion or drawing, the analysis must examine the process from the instant when the workpiece first enters the dies or roll gap until such time as the spatial distributions of the process parameters are unchanging. This may require a large number of increments, and if so, the length of the workpiece which needs to pass between the rolls or through the dies would also be large. The FE analysis of a steady-state process can therefore require considerable computational effort. However, the effort can be reduced. At any instant, there may be large parts of the workpiece which are not affected by the deformation, either having passed completely through the region of deformation or having not yet reached it. Thus only the part of the workpiece actually undergoing deformation needs to be modelled in the FE analysis. Naturally, this part of the workpiece will change during the deformation and so will need to be constantly updated. One way of doing this is to produce an FE mesh that encompasses the entire length of the workpiece and to include an algorithm in the FE program that assembles and solves the stiffness matrices of only those elements in, or just adjacent to, the deforming region (figure 6.19).The nodes elsewhere in the mesh can be assigned default values of displacement if required.The method of selecting elements may be based on an examination of the deformation occuring in the mesh, or may use some previously-determined criterion for gauging the extent of the deformation region. A slightly more complicated procedure, but one which reduces the amount of computer storage required, is to use an FE mesh that models only the part of the workpiece immediately surrounding the deformation region. As the deformation proceeds, a type of re-meshing can be carried out whereby elements are removed from the downstream end of the workpiece and new ones are created at the upstream end (figure 6.20).
112
Implementation of the finite-strain formulation
6.6.3
Three-dimensional analyses
Unless otherwise stated, the principles outlined in this chapter, and the previous one, are applicable equally to those FE analyses which are intended to study general 3-D flow problems, and those which take advantage of the simplifications possible when a metalforming process involves only 2-D flow. The simplifications mean that the element stiffness equations in 2-D programs have fewer degrees of freedom and are much simpler to construct. In addition, a full 3-D analysis requires a mesh with many more elements than are necessary for a comparable 2-D analysis because elements are required throughout the volume. These considerations mean that the space needed to store the global stiffness matrix and associated arrays is very much larger in 3-D programs than in 2-D ones. Not surprisingly, it also takes considerably longer to assemble and solve the 3-D matrix equations. Depending upon the particular computer used to run the FE program, the large storage requirements of 3-D programs may mean that special techniques have to be used. For example, on some machines it may be impossible to store the entire stiffness matrix in main memory, even taking into account the sparse and symmetric nature of this matrix. The solution here is to combine the processes of assembly and conversion to upper-triangular form so that equations can be transferred to secondary disc storage as soon as they no longer play any part in the triangularisation procedure. The equations may then be read back from disc during the back-substitution phase of the solution. Using this frontal-solution technique [6.3], only a small part of the stiffness matrix needs to be in main memory at any instant (figure 6.21). Fig. 6.20 FE grid (a) before, and (b) after re-meshing and re-numbering during an analysis of steady-state rolling.
/
4 3 2 1
— ——
(b)
"^A \ \ \ \ \ \ -Zl \ \ \ \ \ \ \ —TA—\ \ \ \—
1 \
\
\ \ \ \
6.6 Special techniques
113
The disadvantage of the frontal-solution technique, apart from the added complexity, is the large amount of time spent reading from, and writing to disc storage devices. Many modern mainframe computers have what is called a virtual memory. Often the size of arrays permitted under such systems can be essentially unlimited. The main memory of these computers is not, of course, unlimited, and may indeed be quite small. The apparent size of the virtual memory is a result of the operating system using disc devices as a secondary store, just as the frontal technique does. The difference is that a computer with virtual memory moves blocks of information back and forth between main memory and the disc devices quite automatically, a process called paging, so that the computer user is never aware of the fact. The problems of storage will probably not arise in these cases, although the computer operating system may require that special action is taken when using very large arrays. Obviously, a great deal of effort is expended by the computer manufacturers to ensure that transfer of information is performed as efficiently as possible, but it is nevertheless important to remember that, even on virtual machines, 3-D Fig. 6.21 Diagrammatic representation of frontal-solution technique (shaded regions represent information stored in computer memory): (a) equations relating to node I fully assembled, (b) equations relating to node I removed to disc storage.
Ad,
Ad7 s
jAf7
i
I Af7
=
114
Implementation of the finite-strain formulation
FE programs will probably spend a great deal of their time waiting for disc read or write operations. It should also be remembered that although the operating system performs paging as efficiently as it can, it cannot determine with any certainty when it is selecting a page of information for transfer to disc, whether that page will be needed in main memory again quite soon. It is possible, therefore, for pages to be transferred to and from the disc devices quite unnecessarily. Thus while the explicit disc read and write operations used by the frontal technique will not be performed as efficiently as those carried out directly by the operating system, the FE program exploits the nature of the calculation being performed and only writes the information to disc when necessary. It is possible therefore that it is more efficient to use the frontal technique, even on computers with virtual memory. This is a question that can only be decided, on a particular machine, by comparing actual program timings. Whichever method is employed, and whatever the machine used, 3-D FE analyses take a long time to compute. This may or may not be a problem. Dedicated machines can be left running for the number of hours, days, perhaps weeks, required for the analysis. The only problem then is one of machine reliability. On time-sharing systems, there may be a limit to the length of time taken by any one job submitted to the computer. The length of time will frequently be shorter than that required for a complete 3-D FE analysis of a metalforming process. In such situations, it will be necessary to divide the analysis up into a sequence of jobs, each performing a specified number of increments of the calculation. At the end of each stage of the analysis, the program must save in a file all the information that will be required in order to enable the next job to continue the analysis from where it was halted. This technique could also be useful even when using a dedicated computer to avoid the loss of interim solutions. In this case, the FE program would save the restart-information at regular intervals during the analysis. If a malfunction occurs, the analysis could be re-started from the last file of information saved.
References [6.1] Lu, S.C-Y. A consultative expert system for finite element modelling of strip drawing. Proc. 13th Nth. American Manufacturing Res. Conf., SME, pp. 433-41 (1985). [6.2] Mallett, R.L. Finite-element selection for finite deformation elastic-plastic analysis. SUDAM rep. no. 80-4, Stanford University (1980). [6.3] Pillinger, I. The prediction of metal flow and properties in three-
References
[6.4] [6.5] [6.6] [6.7] [6.8] [6.9] [6.10] [6.11] [6.12] [6.13]
[6.14] [6.15] [6.16]
115
dimensional forgings using the finite-element method. Ph.D. thesis, University of Birmingham, UK (1984) (unpublished). Male, A.T. The friction of metals undergoing plastic deformation at elevated temperature. Ph.D. thesis, University of Birmingham, UK (1962) (unpublished). Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Friction in finite-element analyses of metalforming processes. Int. J. Mech. Sci. 21,301-11 (1979). Zienkiewicz, O.C., Valliappan, S. and King, I.P. Elasto-plastic solutions of engineering problems 'Initial Stress', finite element approach. Int. I. Num. Meth. Eng. 1, 75-100 (1969). Yamada, Y., Yoshimura, N. and Sakurai, T. Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by the finite element method. Int. J. Mech. Sci. 10, 343-54 (1968). Nagtegaal, J.C. and de Jong, J.E. Some computational aspects of elasticplastic large strain analysis. Computational Methods in Nonlinear Mechanics, ed. J.T. Oden, North-Holland, pp. 303-39 (1980). Rice, J.R. and Tracey, D.M. Computational fracture mechanics. Numerical and Computer Methods in Structural Mechanics, ed. A.R. Robinson and W.C. Schnobrich, Academic Press, pp. 585-623 (1973). Tracey, D.M. Finite element solutions for crack-tip behaviour in smallscale yielding. Trans. ASME, J. Eng. Mater. Techn.. 98, 146-51 (1976). Alexander, J.M. and Price, J.W.H. Finite element analysis of hot metal forming. Proc. 18th Int. Machine Tool Des. Res. Con/., ed. J.M.Alexander, MacMillan, pp. 267-74 (1977). Liu, C. Modelling of strip and slab rolling using an elastic-plastic finiteelement method. Ph.D. thesis, University of Birmingham, UK (1985) (unpublished). Mahrenholtz, O., Westerling, C. and Dung, N.L. Thermomechanical analysis of metal forming processes through the combined FEM/FDM. Proc. 1st Int. Workshop on the Simulation of Metal Forming Processes by the Finite Element Method (SIMOP-I), ed. K. Lange, Springer, pp. 19-49 (1986). Rebelo, N. and Kobayashi, S. A coupled analysis of viscoplastic deformation and heat transfer. Int. J. Mech. Sci. 22, 699-705 & 707-18 (1980). Clift, S.E., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Fracture initiation in plane-strain forging. Proc. 25th Int. Machine Tool Des. Res. Conf. pp. 413-19 (1985). Al-Sened, A.A.K., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Forming sequences in axi-symmetric cold-forging. Proc. 12th Nth. American Manufacturing Res. Conf, SME, pp. 151-8 (1984).
7
Practical applications
7.1
INTRODUCTION
This chapter will examine the application of FE metalforming techniques to a wide range of industrially-relevant processes. The main reasons for conducting computer simulations of metalforming processes are to: (i) reduce development lead times by minimising the number of experimental trials required (get closer to 'right first time') (ii) reduce development costs, particularly those incurred by the manufacture of expensive dies for experimental trials Both of these considerations result in increased industrial competitiveness and flexibility through the ability to introduce new products quickly and cheaply. Estimates vary of the proportion of forged parts in the UK that have axisymmetric geometries, but this figure probably lies somewhere in the region of 60 to 70%. The development of tooling for axi-symmetric parts does not present anywhere near the difficulties that are associated with the design of dies for non-symmetric components, but even so, computer simulation can significantly speed up the design process for axi-symmetric parts. The savings in time and money will be even greater when non-symmetric parts are to be formed. The emphasis here will therefore be on those aspects of the results that are most important in an industrial context, such as: (i) the prediction of operational parameters (force/stroke/time history, die stressing) (ii) the prediction of product properties (distribution of hardness, residual stresses) and defects (die-filling problems, folds, ductile fracture) (iii) the implications for the design of dies, and of sequences of dies in multi-stage processes The practical considerations (difficulties) involved in the simulations (modelling of boundary conditions, length of computation) will also be discussed.
7.2 Forging
117
The examples in this chapter are set out according to the type of metalforming process involved. The main types of process that will be examined are forging,, extrusion and rolling. Where appropriate, the FE results will be compared with the predictions of simple metalforming theory and experimental tests. 7.2
FORGING
7.2.1
Simple upsetting
Simple upsetting (also known as dumping or cheesing) of a cropped billet is frequently performed prior to more complex forging operations in order to increase the width of the workpiece, and to induce a certain level of plastic work. Simple compression is also widely used in the laboratory, along with the common tensile test, in order to obtain stress/strain data. Whereas the simple upsetting of upright cylinders (and billets under planestrain conditions) is relatively well understood, the compression of workpieces under more complex forming geometries has received less attention. It is in these situations, in which the simple empirical guidelines and rules of thumb derived for axi-symmetric and plane-strain examples no longer apply, that finite-element techniques can be of most value, providing as they do information unobtainable by any other means. As a first example of this, we can consider block compression. Figure 7.1 shows the initial shape of the FE mesh used to model the simple upsetting of a rectangular block of commercially-pure aluminium with high interfacial friction [7.1]. Fig. 7.1 Upsetting of rectangular block - finite-element discretisation.
20 mm
118
Practical applications
432 eight-node iso-parametric 3-D elements were used in the FE mesh. Due to the symmetry, this represented one eighth of the workpiece. The elements were arranged in a simple rectangular configuration. The FE analysis used the finite-strain 3-D elastic-plastic formulation described in Chapters 5 and 6 of this monograph. The material properties of commerciallypure aluminium were used in the calculation, with the yield stress increasing with the amount of plastic strain. The analysis assumed isothermal conditions and a slow forming speed, so strain-rate effects could be ignored. The frictional conditions were modelled by means of the beta-stiffness technique using a value of m = 0.7. For most of the analysis, the height of the mesh was reduced by 2% of its original value during each increment. During the early stages, while there were regions that were still elastic, the program automatically selected smaller incremental deformation steps as described in Chapter 6. The analysis was stopped when the height of the mesh had been Fig. 7.2 Predictions of shape of finite-element mesh at various stage during upsetting: (a) whole mesh; (b) horizontal section through centre of workpiece; (c) longitudinal vertical section through centre.
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7.2 Forging
119
reduced to 40% of its original value, which required 39 increments. Each increment took 190 seconds of CPU time on a CDC 7600 computer to calculate. Figure 7.2 shows the distorted FE mesh and two centre-line sections at various stages during the deformation. The roll-round of the billet onto the dies is clearly seen. This figure also illustrates the formation of concavities on the vertical faces of the workpiece near the external corner, and the associated 'earing' of this corner. This feature is also observed experimentally. The prediction of this sort of detail would be extremely difficult by any other theoretical technique. In figure 7.3, the FE predicted profiles of horizontal and transverse sections through the workpiece are compared with experimentally measured shapes. Agreement between the two is quite close. The discrepancies are partly due to the coarseness of the FE mesh, and partly due to using a frictional restraint in the FE simulation that was apparently larger than that in the laboratory trials. This can be seen from figure 7.3b, where the arrow indicates the current position of the original corner of the experimental workpiece at this stage of the deforFig. 7.3 Comparison of finite-element and experimental profiles at 40% reduction in height: (a) one quarter of horizontal section through centre of workpiece; (b) one quarter of transverse vertical section through centre. exptl. profile
(a)
(b)
120
Practical applications Fig. 7.4 Comparison of finite-element and experimental distributions of Vickers Pyramid Hardness Number (VPN) across one quarter of the transverse vertical section through the centre of the workpiece at 40% reduction in height.
\ ,
finite-element
7.2
Forging
121
mation. Clearly, the amount of interfacial sliding was greater in the experiment than in the FE analysis. The effect of a greater frictional restraint was for there to be greater roll-round of material in the FE case. Despite differences in the frictional conditions, the FE predictions of the internal patterns of deformation agree very well with experimental observations. This is illustrated by figure 7.4, which compares FE and experimental distributions of Vickers Pyramid Hardness Number (VPN) on the transverse vertical centreline section through the workpiece at about 40% reduction in height. (For convenience of presentation, the deformed shapes of the sections have been mapped onto a square base.) The FE hardness values were obtained from the predicted values of yield stress Fusing the relationship VPN — 0.3Y. The overall pattern and level of the distribution is very similar in the two cases, except for the prediction by the FE program of a peak value of hardness on the top face of the billet near the edge. This is again a result of the differences in the frictional conditions and the larger amount of roll-round calculated by the FE analysis. The predicted distribution of pressure across one quarter of the top of the billet at 40% reduction in height is given in figure 7.5. This shows that the pressures decreases towards the centre of the face. Experimental work carried out by Nagamatsu andTakuma [7.2] has also provided evidence for the formation of a 'friction valley' in the upsetting of billets with this type of geometry. The finite-element analysis calculated an overall value of 200 kN for the deforming load at this stage of the process. This compares with a value of 210 kN measured experimentally. Fig. 7.5 Distribution of pressure (P) across one quarter of top surface of finite-element mesh at 40% reduction in height.
r300
122
Practical applications
7.2.2
Upset forging
Upset forging is commonly used to make axi-symmetric components from a bar. A typical component is shown in figure 7.6. Fig. 7.6 Geometry of component produced by upset forging.
26.5
40
The FE analysis of the forging of this component performed by Eames et al. [7.3] again used the finite-strain 3-D elastic-plastic formulation described in Chapters 5 and 6. The forging was performed slowly under isothermal conditions. The yield stress of the material (aluminium alloy HE30TF) varied only with strain. The beta-stiffness technique was used to model the frictional conditions between the dies and the billet with m = 0.1 to simulate lanolin lubricant. The predicted deformation is shown in figure 7.7b. This analysis was also performed with m = 0.7, representing no lubrication. The forces for both conditions are shown in figure 7.7a. Since this example is axi-symmetric, only a single-layer of 145 eight-noded iso-parametric hexahedral elements needed to be used in the analyses. These were arranged to form a thin wedge. The nodes on the outer faces of this wedge were constrained to move on radial planes to maintain the conditions of axial symmetry. The lower part of the mesh was contained within a closely fitting cylindrical die, while the upper part was located within a conical die. In this process there is very little sliding contact between the workpiece and the dies, the effect of the upsetting being for the material to roll onto the inside of the conical die. As a result, there is little difference between the forces (both experimental and FE) obtained for the two levels of friction. It can be seen that the FE results are in reasonable agreement with the experimental values.
7.2.3
Heading
In the heading process, a portion of a bar is held between clamps or grippers and the exposed portion is compressed by a flat or shaped punch to spread the material into some sort of cap attached to the undeformed stem. It is used to make a very wide variety of components, including fastenings, rivets and bolts. Heading can of course be one part of a multi-stage forming process.
7.2
Forging
123
Figure 7.8 shows the finite-element mesh used by Hussin et al. [7.4] to model an example of a heading operation, in which a cylindrical bar of radius 25.4 mm and length 50.8 mm is gripped for half its length and the remaining portion upset by a flat punch. The FE analysis used a small-strain 2-D (axi-symmetric) elasticplastic formulation. The calculation assumed isothermal conditions and no strainrate effects, the yield stress of the commercially-pure aluminium varying only with plastic strain. The beta-stiffness technique was used to model the frictional conditions between the punch and the workpiece, and two analyses were performed, one using a value of friction factor m = 0.1, the other using m = 0.7. These two levels of friction were intended to correspond to heading with a good lubricant (such as lanolin) and heading with no lubricant at all. In both analyses, 105 eight-noded iso-parametric quadrilateral elements were used to model half the Fig. 7.7 (a) Comparison of finite-element and experimental forging forces (kN) during upset forging: • experimental high friction; finite-element high friction; • experimental low friction; -— finite-element low friction. (b) Comparison of distorted finite-element meshes and experimental profiles at various stages in upset forging process with lanolin lubricant. 1000
Force (kN) 500
20
(b)
\
Deformation
124
Practical applications
section of the workpiece, arranged as in figure 7.8. Larger elements were used in the gripper region of the mesh because it was expected that little or no deformation would occur here. During the FE analysis, the exposed portion of the mesh was reduced by 1% of its original height during each increment of the calculation up to a maximum of 60%. The FE calculation was performed on a Future Computer FX30 described in Chapter 4. Figure 7.9 shows the distorted FE meshes, drawn for the full cross-section, for the two levels of friction, at intervals of 10% reduction in height. The differences in the pattern of deformation between the two frictional conditions can be clearly seen. It can also be seen that in the later stages of the deformation, the FE program predicts deformation within the gripper region of the mesh. This is not observed experimentally, indicating that a finer mesh should have been used to model the exposed part of the workpiece and at least the upper part of the gripper region. Remeshing in this region at various stages of the simulation would also have helped to reduce this effect. Figure 7.10 compares FE predictions of Vickers Pyramid Hardness Number with experimentally-measured values for the two levels of friction at a deformation of 47% reduction in height. The FE values were obtained from the distribution of yield stress as described in section 7.2.1. Although the FE and experimental values differ by about 5-10% over most of the section, the overall pattern is similar. There is a larger discrepancy between the two sets of values in the lower part of the head portion. This is again a result of a coarse mesh and the FE Fig. 7.8 Heading - initial finite-element mesh. I
Punch
I
Billet
Grippers
7.2
Forging
125
program predicting that the region of deformation extends down into the gripped part of the mesh. Although the patterns of internal deformation obtained with the two levels of friction are strikingly different, the forming loads in the two cases are very similar. Figure 7.11 compares the forces measured experimentally with the FE results. Agreement is very good. The kink in the FE-predicted load/deformation curve for m = 0.7 at 42% reduction is due to the folding round of an element onto the punch surface.
7.2.4
Plane-strain side-pressing
Plane-strain side-pressing is the name given to the transverse forging of billets Fig. 7.9 Finite-element simulation of heading - FE meshes between initial state and 60% reduction in height of head at intervals of 10%: (a) m = 0.1; (b) m = 0.7.
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126
Practical applications
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Fig. 7.10 Comparison of finite-element and experimental values of Vickers Pyramid Hardness Number (VPN) across section of billet during heading at 47% reduction in height: (a) m = 0.1; (b) m = 0.7.
Fig. 7.11 Comparison of finite-element and experimental results for the variation of forging force (kN) during heading operation as a function of reduction in height of head: -#- experimental (m = 0.1); • experimental (m = 0.7); prediction.
10
20
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40 50 60 7c Deformation
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7.2 Forging
127
in which no deformation is assumed to occur in the longitudinal direction. A typical example would be in the early stage of the forging of a circular bar into a turbine blade. Here, friction between the shaped dies and the workpiece ensures that there is little increase in the length of the billet. The process is discussed here principally in order to illustrate the ability of finite-element techniques to predict when and where the workpiece is likely to fracture during plastic deformation. This ability is of great importance, particularly when materials of low ductility, such as those chosen for aerospace applications, are used. Three geometries of side-pressing between flat dies were examined by Clift et al. [7.5]. In the first of these, the workpiece had a circular cross-section with diameter 20 mm. In the other two cases, parallel flats were machined on the billets before forging. For one of these specimens, the ratio ///Wof the machined height of the billet to the width of the machine flat was 2.03, and for the other specimen the ratio H/W was 1.33 (figure 7.12). The large-displacement 3-D elastic-plastic formulation described in Chapters 5 and 6 was used for the FE analysis. The program was constrained to model plane-strain behaviour by preventing the movement of nodes in the longitudinal direction. The forging was assumed to be carried out slowly under isothermal conditions, so there were no strain-rate or temperature effects in the model. The yield stress was calculated as a function of plastic strain in order to simulate the work-hardening properties of 7075 aluminium. This fairly brittle material was chosen for the trials so that fracture would be observed at an early stage of the deformation. The beta-stiffness technique was used to model the frictional conditions between the workpieces and the dies, a value of friction factor m = 0.25 being selected to simulate unlubricated dies. Approximately 80 eight-node iso-parametric hexahedral elements were used to model one quarter of the cross-section of the three types of workpiece. The initial FE meshes are shown in figure 7.12. Because these were plane-strain examples, only a single layer of elements was required in each case. Figure 7.13 shows the sites of fracture initiation observed experimentally for the three geometries of specimen considered here. The specimen that was originally circular started to fracture in the centre of the workpiece at approximately 16% reduction in height. In contrast, fracture in the specimens with the machined flats started at the outer surface of the billets, at points that were near the original positions of the corners of the specimens. The levels of deformation for fracture initiation in these two cases were 18% for the specimen with H/W= 2.03 and 16% for the specimen with H/W= 1.33. Figure 7.14 shows the FE meshes for the three geometries at the levels of deformation at which fracture started in each case. The prediction of fracture requires the monitoring of some fracture-initiation
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Fig. 7.12 Three geometries of plane-strain side-pressing showing initial finiteelement meshes: (a) full circular section; (b) circular section with machined flats, ratio of height of billet H to width of flats W = 2.03; (c) circular section with machined flats, H/W = 1.33. The material used is 7075 Al.
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7.2 Forging
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Fig. 7.13 Site of fracture initiation in experimental plane-strain side-pressing specimens: (a) initially circular section; (b) initial H/W = 2.03; (c) initial H/W = 1.33. (a)
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Practical applications
function at all points of a body, and determining where and at what level of deformation the value of this function exceeds some critical empirical quantity. Many different functions have been proposed [7.6], most of which depend upon the state of stress, and some of which involve integration along the strain paths of points within the body. Since FE techniques produce highly detailed information about the distribution of stress, strain and other quantities throughout the deformation, they are ideally suited to the task of predicting the onset of fracture. One fracture-initiation measure that has proved to be particularly successful is called the generalised plastic work criterion. This integrates the plastic work per unit volume from the start of the deformation up to the current stage: Total generalised plastic work
-i
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(7.1)
per unit volume The values of total generalised plastic work per unit volume are plotted in figure 7.15 as a function of deformation for the three cases examined. The curves Fig. 7.15 Variation of the maximum value of total generalised plastic work per unit volume up to the levels of deformation at which fracture initiated experimentally for the three geometries of plane-strain side-pressing: (1) maximum value for H/W = 2.03, node 191; (2) maximum value for H/W = 1.33, node 169; (3) maximum value for circular section, node 1. Location of nodes given in figure 7.14.
5
10 Deformation %
7.2
Forging
131
drawn are for the nodes in the FE meshes at which this quantity has its greatest values during the deformation. The position of these nodes is indicated in figure 7.14. It can be seen that the FE program predicts that the maximum values of this fracture function occur at the positions where fracture is observed to start in the laboratory trials. Moreover, the curves in figure 7.15 attain the critical value for the onset of fracture at levels of deformation which are quite close to the values measured in the experimental trials. Further experimental studies [7.7] have shown transition from centre cracking to corner cracking at a value of H/Wapproximately equal to 2.5. This transition value is also predicted by FE analyses using the generalised plastic work criterion.
7.2.5
Rim-disc forging
Rim-disc forging is a form of backward extrusion in which a punch is pushed into the centre of a workpiece held within a container to form a rim around a central depression. It differs from backward extrusion (considered later) in that the deformation is non-steady state throughout. A small-strain axi-symmetric (2-D) elastic-plastic FE program was used to study this process [7.8]. The strain-hardening properties of commercially-pure aluminium were modelled and the deformation was assumed to be performed slowly at room temperature. The beta-stiffness friction technique was used to simulate the frictional conditions between the billet and the die and punch. A value of friction factor m = 0.04 was selected to model good lubrication. Fig. 7.16 Initial finite-element mesh for rim-disc forging.
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132
Practical applications
The FE mesh represented one half of the cross-section of the workpiece which had a diameter of 39 mm and a height of 13.3 mm (figure 7.16). A regular arrangement of 546 constant strain triangular elements were used for the analysis which was performed with increments of 0.25% reduction in height of the central portion of the billet. Figure 7.17 shows the distorted FE meshes at various stages of the deformation. The diagonal band of high shear can be seen in figure 7.17d leading from the corner of the punch downwards at an angle of approximately 45°.This forms the boundary of a conical region of lightly deformed material beneath the punch. The pattern of deformation agrees broadly with experimental observations of gridded specimens, though the shape of the extruded free surface shows the effect of the restraint to deformation imposed by the sharp corner of the punch in the FE simulation. The pattern of flow in the FE analysis can be improved by re-meshing at intervals during the deformation. An example of this is given in reference [7.9] and illustrated in figure 7.18. The distributions of generalised strain predicted by the FE program are shown in figure 7.19. This indicates a region of low deformation at the bottom of the billet a short distance away from the corner in addition to that directly under the punch. Examination of the grain structure on etched surfaces of sectioned workpieces confirms these predictions. The 'dead zone' near the bottom corner Fig. 7.17 Finite-element predictions of internal distortion during rim-disc forging: (a) 10% reduction in height of central portion; (b) 16% reduction; (c) 21% reduction; (d) 26% reduction.
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Fig. 7.18 Internal deformation in backward extrusion predicted by the small deformation program, with m = 0.6, at each 5% deformation up to 60%. The results were obtained using re-meshing and the supergrid technique.
133
134
Practical applications
of the workpiece has sometimes been found to be the site of defects in industrial components of this type. The comparison of the FE and experimental distributions of VPN hardness illustrated in figure 7.20 again shows that the FE program predicted the correct pattern of deformation, though at some points it over-estimated the values of strain. This is almost certainly because of the inability of the FE model to simulate correctly the flow around the corner of the punch which has led to excessive levels of deformation in the body of the workpiece. Fig. 7.19 Distribution of generalised strain in rim-disc forging predicted by finite-element analysis at 26% reduction: (A) 0.61-1.03; (B) 0.46-0.60; (C) 0.31-0.45; (D) 0.16-0.30; (E) 0.11-0.15.
Fig 7.20 Comparison of finite-element predicted values and experimentallymeasured values (in parentheses) of VPN hardness in rim-disc forging at 26% reduction. ^~
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/ 44(43) 38(35) 39(34) 1 A(\( 43) 37(30) 36(28) 36(29) 36(31) 36(32) 49(45) 45(44) 40(39) 40(34) 46(41)44(43)41(45) 37(31) 37(30) 37(32) 39(36) 44(39) 44(41)42(41) 40(40) 40(37) 39(35) 40(34) 40(34) 43(40) 44(37) 40(37) 40(39) 39(39) 39(39) 42(39) 42(38) 43(37) 42(38) 40(39) 38(39) 37(34) 38(38) 38(41) 44(40) 44(41) 42(40) 40(39)38(38) 36(34) 35(34) 36(35) 38(40)
7.2
7.2.6
Forging
135
Extrusion-forging
Extrusion-forging, as its name implies, embodies two forms of deformation in one process. This combination of different types of deformation is typical of many operations and is illustrated in an idealised form in figure 7.21. In this example, a billet is compressed between a flat platen and a platen with a central orifice. It is useful because by varying certain geometric parameters, the relative importance of the two deformation mechanisms, extrusion and forging, can be altered. The results shown here were obtained by Kato et al. [7.10, 7.11] and illustrate the influence of the initial geometry of the billet and the size of the central orifice. The finite-strain 3-D program was used for these analyses, constrained to reproduce plane-strain deformation. Sticking friction was assumed on both Fig. 7.21 Idealised illustration of flow in extrusion forging: (a) before deformation; (b) during deformation.
(a)
(b)
136
Practical applications
platens and only a small amount of deformation was undertaken in order to examine how the flow patterns developed before significant geometric changes occurred. Commercially-pure aluminium was used as the model material. Rate and temperature effects were not included. The results of the analyses showed that three distinct modes of flow could be identified. Mode I, figure 7.22a, is typified by a rigid-core region around the plane of symmetry. This is predominantly a forging operation. Mode II, figure 7.22b, clearly shows mixed deformation as the forging of material between the platens is accompanied by extrusion into the central orifice. A neutral point, indicating a flow divide, can be seen along the upper surface in contact with the platen. Mode III, figure 7.22c, is very similar to simple upsetting since material Fig. 7.22 Typical flow patterns for (a) mode I, (b) mode II and (c) mode III types of deformation.
mode I b/a = 2, h/a = 4
(a)
mode II b/a = 8, h/a = 4
(b)
1
nil
t i l l I 1 I IV i 11 I I I \ \ v H i l l \ \ \ N> » I V\
(c)
\
\
\
\
mode III b/a = 4, h/a = 8
7.2
Forging
137
h/a
b/a
Fig. 7.23 Flow pattern 'classification' diagram. beneath the orifice moves down towards the lower platen along with the rest of the material. It is possible to produce a 'classification' diagram as shown in figure 7.23, which relates modes of flow to initial geometry. This is somewhat of a simplification, as the boundaries of the regions in the diagram are not actually as distinct as they are shown to be.
7.2.7
4
H'-section forging
Full 3-D analyses require many elements and take a correspondingly large amount of computer time to perform, but it is sometimes possible to use a two-dimensional simplification to examine a particular feature or stage of a complex 3-D forging. Fig. 7.24 Finite-element mesh and initial positions of die boundary surfaces for 'H'-section forging.
138
Practical applications
Figure 7.24 shows an example of this. This depicts an FE mesh modelling one half of an 'H'-section of a long bar. This represents one stage in a sequence of operations, so there is already some flash on the pre-form (to the right in the figure). During the forging of this component it is found that there is very little change in the length of the bar, so the behaviour of the central portion can be modelled satisfactorily assuming plane-strain conditions. The finite-strain 3-D elastic-plastic formulation described earlier in this monograph was used to perform the analysis. The 336 eight-node iso-parametric hexahedral elements were arranged in a single layer as shown in figure 7.24. The nodes were prevented from moving perpendicularly to the plane of this figure in order to model plane-strain deformation. Because of the offset parting line, the 'H'-section possessed only one-fold symmetry, and so the FE mesh represented one half of the section. The initial positions of the dies are also shown in figure 7.24. These were modelled in the FE program by means of 18 primitive boundary surfaces. A friction factor of m = 0.2 was assumed between the dies and the billet, the frictional restraint being imposed in the FE calculation by the beta-stiffness technique. During each increment of the analysis, the upper boundary surfaces were moved downwards in order to reduce the thickness of the central part of the section by a nominal value of 1.05% of its original height. The time step for the increment was chosen to give a nominal strain rate of 1.0 s"1 in this region. The model material was Al 1100, the yield stress of which depended upon plastic strain, strain rate and temperature. The following thermal properties were used in the analysis: thermal conductivity = 242 J/msK thermal capacity/volume = 2 . 4 3 MJ/m 3 K die-interface conductivity = 35 kJ/m 2 K Since this was a warm forging, the billet was assumed to be at a temperature of 700 K and the dies at a temperature of 600 K at the start of the deformation. The complete FE analysis of this deformation up to a reduction of 26% in the height of the central portion of the section required approximately five hours of CPU time on a Honeywell DPS8 mainframe computer. Figure 7.25 shows how the plastic region develops during the early stages of the deformation. At first, the billet is only in contact with the dies in the regions A, G-I and L marked in figure 7.24, so the initial deformation is principally that of simple compression of the central part of the section. The shear bands can be seen clearly in figure 7.25a. Gradually, the region of plastic deformation grows until at 2% reduction it extends right across the section and into the flash region (F, G). Even at this stage, however, the upper and lower lobes of the section are still deforming elastically.
7.2
Forging
139
Fig. 725 Development of plastic zone (shaded) during early stage of 'FTsection forging: (a) 0.05% reduction; (b) 0.065%; (c) 0.2%; (d) 0,5%; (e) 1%; (f) 2%.
(a)
(b)
(c)
140
Practical applications
Fig. 7.25 (continued)
(d)
(e)
(f)
7.2 Forging
141
Fig. 7.26 Flow velocity vectors at various stages in forging of 'H'-section: (a) 2% reduction; (b) 3%; (c) 10%; (d) 12%; (e) 15%; (f) 26%.
(a)
(b)
(c)
142
Practical applications
Fig. 7.26 (continued)
(d)
(e)
7.2
Forging
143
T h e velocity vectors in figure 7.26 illustrate the patterns of flow in the ' H ' section in more detail. At 2 % reduction for instance (figure 7.26a), it can be seen that the u p p e r lobe is being moved across by the outward flow of material from the central region, whereas the lower is prevented from moving by its contact with die surfaces H and / . T h e r e is therefore very little flow of material in this part of the workpiece. At about 3 % reduction (figure 7.26b), the upper lobe of the 'H'-section comes into contact with the die at surface D and its outward movement is halted. Material from b e n e a t h the central indenter now flows mainly into the bottom lobe, and begins to fill u p the space near surface K. There is also a small amount of flow into the u p p e r lobe. This situation persists until about 10% reduction (figure 7.26c), by which time the downward movement of die surface D has begun to reverse the flow in the upper lobe. Material is still filling the space in the b o t t o m , but is now also flowing into the flash region. This behaviour is somewhat intensified at 12% reduction (figure 7.26d), but by 15% reduction in height (figure 7.26e), the situation has changed again. At this stage, the u p p e r die is also in contact with the workpiece at surface B, and this has reversed the flow in the upper lobe once again, but this time only partially. T h e right-hand side of the lobe is still being pushed down by the die wall, but material is flowing upwards on the left-hand side. The complex pattern of flow is shown in greater detail in figure 7.27, in which a flow divide on the right-hand side can be seen clearly, together with a rotational pattern on the left-hand side. Finally, figure 7.26f shows that at the end of the deformation, the lower lobe has completely filled, whereas there is still some flow of material into the gap at the t o p . Most of the flow however is into the flash. T h e distribution of plastic strain at the end of the deformation is illustrated in figure 7.28. This shows how most of the deformation has occurred in an approximately horizontal band across the section; there is relatively little strain in the two lobes. Much of their movement has occurred as almost rigid-body m o v e m e n t . T h e region of highest strain is near the lower right-hand corner of the die (surface H). T h e t e m p e r a t u r e contours at 2 2 % reduction shown in figure 7.29 indicate that the principal determinant of temperature in this forging example is the effect of die chilling on the workpiece. T h e hottest part of the workpiece is half-way between the lower and upper lobes, but the temperature gradient between the centre and the outside of the billet is not uniform. Those parts of the billet that have been in contact with the dies the longest time (e.g. A, D, G-I and L ) show the greatest a m o u n t of cooling. A n interesting consequence of this is that the t e m p e r a t u r e difference across the flash thickness has led to a difference in the flow stress, and to an asymmetrical flow of material into the flash gutter.
144
Practical applications
Fig. 7.27 Close-up of flow velocity vectors in upper lobe of 'H'-section forging at 15% reduction. Fig. 7.28 Contours of generalised plastic strain in 'H'-section forging at 26% reduction.
0.0
1.0
2.0
X10' 1 8.0
9.0
3.0
4.0
5.0
6.0
7.0
8.0
7.2
Forging
145
Absolute temperature contours (K) 62.3 62.5 62.8 63.0 63.3 63.5 63.8 64.0 64.3
X101
...222
64.3 64.5 64.8
Fig. 7.29 Contours of temperature (K) in 'H'-section forging at 22% reduction.
7.2.8
Forging of a connecting rod
Industrially, connecting rods are usually forged from steel at a temperature of approximately 1200 °C.The forging is typically carried out in six stages, starting from the original bar stock and ending with a coining operation before the final clipping of the flash. The finite-element analyses described here model a laboratory simplification of one of these stages (from final preform to mould forging) using commercially-pure aluminium deformed at room temperature [7.12]. The initial preform geometry and the desired shape of the billet at the end of this stage of the deformation are shown in figure 7.30.The symmetry of the component meant that only one quarter of the preform needed to be modelled. The initial FE mesh is shown in figure 7.31. This contained 600 eight-noded 3-D linear isoparametric elements. The FE analysis was carried out using the 3-D large-displacement elastic-plastic approach described in Chapters 5 and 6 of this monograph. The commercially-
146
Practical applications
pure aluminium used for the model was assumed to deform isothermally and no strain-rate effects were included. Sixteen geometric primitive surfaces were used to model the complex shape of the dies. For simplicity, two complete analyses were performed, one assuming zero-friction conditions between the billet and the dies and the other assuming sticking friction. During each increment of the Fig. 7.30 (a) preform and (b) required final shape of connecting rod forging. -035.5
8.5 (b)
T" 1
1
^ ^
r— 025
si
YZZZ
4.5
8.5
Fig. 7.31 Finite-element idealisation of connecting rod preform.
7.2 Forging
147
analysis, the thickness of the billet at the centre of the big end was reduced by no more than 0.5% of its original value. Each increment took approximately 120 s of CPU time on a CDC 7600 computer, a complete analysis taking about 5y hours. Figure 7.32 compares the experimentally measured change in length of the billet during the deformation with the values predicted by the two FE analyses. The predictions of the sticking-friction and zero-friction analyses are markedly different. Despite the pronounced scatter in the experimental results, it can be seen that the sticking-friction results agree very well with the actual measurements. The patterns of deformation predicted by the two FE analyses are compared in greater detail in figures 7.33-7.35. The vertical sections through the deformed FE meshes (figures 7.33 and 7.34) show that the pattern of deformation in the zero-friction case is much more homogenous than that obtained with sticking friction. The velocity vectors illustrated in figure 7.35 clearly indicate other essential differences in the two modes of flow. Figure 7.35a shows that when there is no frictional restraint, flow on the horizontal centre-line plane has a predominant longitudinal component, so that material extruded from under the big-end indenter tends to elongate the billet. In contrast, with sticking friction, material flows radially outwards from under the big-end indenter, tending to increase the amount of flash around this region. There is practically no longitudinal flow along the connecting arm of the billet. Instead, flow in this region is almost entirely in a transverse direction and results from the formation of the indentation in this arm. However, one feature of the experimental forging that was predicted by the FE analysis with sticking friction was the reduction in the width of the flash at Fig. 7.32 Experimental measurements (A) and finite-element predictions ( ) of changes in length (8L) of connecting rod forging plotted against percentage reduction in height, R.
10
20
30
40
50
60
70
80
148
Practical applications
(a) zero friction
(b) sticking friction I
Fig. 7.33 Transverse vertical sections through big end of deformed FE connecting rod meshes at about 60% reduction assuming (a) zero friction and (b) sticking friction.
Fig. 7.34 Longitudinal vertical sections along centre-line of connecting rod showing FE meshes at about 60% deformation assuming (a) zero friction and (b) sticking friction.
(a) zero friction
(b) sticking friction
7.3 Extrusion
149
(a)
Fig. 7.35 Material velocity vectors on horizontal centre-line section of connecting rod at about 60% deformation predicted by FE analyses assuming (a) zero friction and (b) sticking friction.
the point where the small end joins the connecting arm. This can be clearly seen in figure 7.35 at the location marked. Figure 7.36 compares experimentally measured values of hardness on the vertical plane of symmetry of the connecting rod with the values predicted by the sticking-friction FE analysis. Despite the coarseness of the FE mesh, the two distributions are in good agreement. Fig. 7.36 Comparison of experimental values (top) and sticking-friction finiteelement predictions (bottom) of Vickers Pyramid Hardness distributions on longitudinal vertical plane through the centre-line of the connecting rod. Hardness ranges: (kgf/mm2): a, 23-35; b, 35-45; c, 45-55; d, 55 + . Experimental result
Finite element prediction (sticking friction)
7.3
EXTRUSION
7.3.1
Forward extrusion
Forward extrusion is used to produce a large variety of continuous sections. It may also form part of a more complex forging operation, such as the formation of a central boss in a gear blank. It is in this context that the process is examined here.
150
Practical applications
Figure 7.37 shows the example studied by Hussin etal. A short billet is positioned within a container prior to extrusion through a conical die. The billet is 25.4 mm in diameter and 20.32 mm long. The die orifice has a diameter of 15.86 mm, corresponding to an area reduction of 61% .The included die angle is 124 degrees. A small-strain 2-D (axi-symmetric) elastic-plastic finite-element analysis was used to study this process. The material (commercially-pure aluminium) was assumed to be extruded slowly and under isothermal conditions, with the yield stress depending only upon strain. The beta-stiffness friction model was used, with a value of friction factor m = 0.7 employed to correspond with unlubricated contact between the billet and the container and punch. The initial FE mesh consisted of 109 eight-noded iso-parametric quadrilateral elements arranged in a simple rectangular grid (figure 7.37). Since the billet was symmetrical about the centre line, only half of the cross-section was modelled. During each increment of the analysis, the punch was displaced by 1% of the original length of the billet. Because of the severe deformation occurring in the extrusion process, particularly at the corner of the orifice, the mesh was reformed after every 5% deformation. Information about the internal flow distribution was used to update the nodal co-ordinates of a reference mesh at the end of each increment to indicate the pattern of lines that would be obtained on a gridded experimental specimen (super-grid technique). The distorted FE reference grids are shown at stages throughout the deformation in figure 7.38. The areas of gross deformation at the corners of the orifice can be seen clearly, as can zones of much lower deformation near the internal corners of the container. This is bounded by a diagonal band of shear, making an angle of approximately 45 degrees with the container wall. Fig. 7.37 Initial finite-element mesh for forward extrusion.
32 mm
[ mm.
i Initial billet
o
-15.86 mm-
7.3 Extrusion
151
The values of VPN hardness predicted by the FE program are compared with the experimentally-measured values in figure 7.39. The FE hardness numbers were calculated from the strain values as described in Section 7.2.1. The two sets of values agree quite well and support the evidence of the deformation patterns in that they indicate regions of high deformation at the corner of the orifice, and lower deformation beneath the punch. Figure 7.40 compares the FE and experimental results for the variation of the extrusion force during the deformation. The FE program tended to over-estimate the loads, but the slope of the predicted curve is in excellent agreement with the experimental behaviour. Fig. 7.38 Finite-element predictions of the distortion of an initially-regular grid at various stages of forward extrusion (super-grid technique).
Initial state
18% Deformation
tI1 *•• 5% Deformation
I
23% Deformation
13% Deformation
28% Deformation
152
Practical applications
—
—
—
,
•
^
34.3
34.8
32.4
42.3
43.4
43.8
44.6
43.6
44.4
37.1
36.2
33.7
44.1
44.9
47.3
52.3
49.0
47.3
54.1
47.0
52.3
49.1
51.9
54.5
48.3
49.7
56.0 i
50.3
53.6 58.7
48.6
51.8
53.9 52.1
48.8
51.2
52.3
47.9
52.1
47.9 46.2
45.4
49.0
53.3 48.9
48.7
47.5
45.4
44.4
43.6 42.9
40.9
47.0
46.8 46.0
44.3
41.8
40.1
40.2
38.6 38.6
34.8
42.3
41.7 41.2
40.2
38.7
61.2
55.0 53.2
(a)
50.0
(b)
Fig. 7.39 Comparison of VPN hardness values at 28% reduction of original billet length: (a) finite-element values; (b) experimentally-measured values.
Fig. 7.40 Comparison of the finite-element predictions and measured values of the variation of forging force (kN) during forward extrusion: • experimental; prediction.
180 160 140 120 c 100 80 60 40 20 0 a> o
a
'I
/—*-*—IT-o^"" / / - / -/ -/ / 4. 8. •12 16
20 Deformation %
24
28
30
7.4
7.4
ROLLING
7.4.1
Strip rolling
Rolling
153
In strip rolling, the width of the workpiece is very much greater than its thickness. As a result, the frictional restraint to transverse movement of material is large and there is little sideways spread. Strip rolling can therefore be reasonably modelled as a plane-strain deformation. The FE analyses carried out by Liu et al. [7.13, 7.14] used the finite-strain 3-D elastic-plastic formulation described earlier in this monograph. The rolling was assumed to be carried out under conditions that allowed the effects of strain rate and temperature on the properties of the material to be ignored. The strainhardening characteristics of copper were used in the analysis.The friction between the roll and the strip was modelled by means of the beta-stiffness technique using a value of friction factor m = 0.2. In this example, the strip was 1.56 mm thick and the roll has a diameter of 78 mm. The thickness of the strip was reduced by a nominal value of 22.76% in one pass. Due to symmetry, only one half of the strip needed to be modelled. A strip length of 31.4 mm was used for the analysis to ensure that steady-state conditions would be reached before all the strip has passed between the rolls. This required a total of 88 eight-noded iso-parametric hexahedral elements arranged in a single layer with four elements in the thickness direction. The nodes of the mesh were prevented from moving in the direction of the roll axis in order to specify plane-strain conditions. Fig. 7.41 Frictional boundary conditions during finite-element simulation of rolling.
Friction layer
' Rolling direction
154
Practical applications
Figure 7.41 illustrates how the boundary conditions were imposed on the FE mesh. Any nodes that came into contact with the roll were constrained to move tangentially to it. The adjacent friction-layer nodes were rotated by a small angular amount about the roll axis each increment, which tended to pull the upper layer of the mesh into the roll gap. Note that the velocity of the top of the strip was not prescribed beforehand but was a function of the stiffness of the friction layer and hence of the frictional conditions at the roll surface. On exit from the roll of course, the surface of the strip was moving faster than the roll, and the effect of the friction layer was to retard this movement. The FE program was therefore able to predict the position of the neutral point on the roll. Figure 7.42 shows the deformed FE meshes from just after the strip has entered the roll gap until the steady state. This figure shows clearly that plane sections through the workpiece do not remain plane as they pass under the roll (an assumption of simpler theories [7.15]). Also the curvature of these sections reverses during the rolling, an effect that is more pronounced for higher levels of friction. The velocity vectors, relative to the horizontal velocity of the neutral point of the roll, are shown at various stages in figure 7.43. By subtracting the horizontal Fig. 7.42 Predicted finite-element deformed meshes at stages during the rolling of wide copper strip, (d) is for steady state conditions.
(a)
7.4 Rolling
155
component of the neutral-point velocity in this way, the pattern of flow and the change in position of the neutral surface in the strip can be seen more clearly. Figure 7.44 compares experimental [7.16] and FE results for the distribution of pressure under the roll during steady-state rolling. Two FE curves are shown, one for m = 0.2 and the other for m = 0.4. The lower value of m corresponds to the level of friction estimated for the experimental trials, but it can be seen that the FE results for this level of friction are much lower than the experimental measurements. FE predictions for a friction factor of m = 0.4 agree very well with experiment. Estimating friction factors from experimental tests using pressure pins is very difficult. The results here suggest that the higher value may be more appropriate.
7.4.2
Slab rolling
In slab rolling, the width of the workpiece is of the same order as its height. In such cases, the bulging and spread of material as the workpiece passes through Fig. 7.43 Finite-element flow velocity vectors in rolling of copper strip relative to the horizontal velocity of the neutral point on the roll.
156
Practical applications
Rolling direction 600
0.06 Arc of contact
Fig. 7.44 Distribution of pressure across the contact region during the rolling of wide copper strip: (1) experimental measurements; (2) FE predictions for m = 0.4; (3) FE predictions for m = 0.2.
the roll gap can be quite large and it is very useful to be able to predict spread and the profile of the surface in order to reduce material wastage. The FE treatment of this type of rolling requires a full 3-D approach. The large-displacement 3-D elastic-plastic program described in this monograph was used by Liu et al. [7.17] to examine slab rolling. The deformation was assumed to be carried out at room temperature using mild steel (AISI 1080). Several rectangular sections of slab were studied. The example examined here had a height and width of 25.4 mm (W/H = l).This was subjected to a reduction in thickness of 20% in a single pass between rolls 604.6 mm in diameter. To model one quarter of the slab, 300 eight-noded iso-parametric hexahedral elements were used, arranged as in figure 7.45. The length of the FE mesh was Fig. 7.45 Initial finite-element mesh for slab rolling.
W/H=\
7.4
Rolling
157
estimated at about three times the nominal contact length, which ensured that steady-state conditions could be attained during the analysis. The beta-stiffness technique was used to model the frictional conditions at the roll surface, the friction factor m having a value of 0.5. The slab was drawn into the roll gap by imposing a small rotation to the roll surface each increment. The shapes of various vertical sections through the steady-state FE mesh are shown in figure 7.46. These indicate that the bulge does not develop until the Fig. 7.46 Transverse cross-sections through deformed finite-element mesh for slab rolling.
F
I
I F
A-A
B-B
C-C
D-D
Tin E-E
7T
_B
F-F
158
Practical applications
/ / / / Fig. 7.47 Finite-element flow velocity vectors on horizontal mid-plane section through workpiece relative to the horizontal component of velocity of neutral point of roll surface.
material is well into the roll gap (C-C), and that before this the outer surface of the slab may even be slightly concave. The velocities of points on the mid-plane horizontal section of the workpiece relative to the horizontal component of velocity of the neutral point are illustrated in figure 7.47. This shows that the transverse flow of material is not uniform throughout the deformation zone, but increases from the centre to the edges of the slab. Figure 7.48 compares the FE predictions for the overall spread of the material, Fig. 7.48 Comparison of finite-element predicted values of spread in slab rolling for various reductions with experimental and theoretical results.
5
10 15 20 Reduction in slab thickness %
7.5 Multi-stage processes
159
for a range of reductions in thickness from 5% to 20%, with experimental results [7.16] and previous empirical theory [7.18]. The FE results agree very well with experiment except for small reductions in thickness.
7.5
MULTI-STAGE PROCESSES - FORGING SEQUENCE DESIGN,
7.5.1
Automobile spigot
For economic reasons, spigots, gear-wheel blanks and other such components are frequently forged from comparitively thin bar stock. Because of the limitations of press capacity and the risk of buckling, this cannot be accomplished in a single operation. Figure 7.49 shows the sequence of operations used to forge a typical automobile spigot [7.19]. The cropped or sawn bar is first forced into a set of conical-ended dies to forge a chamfer on the ends (stage 1). This has the effect of tidying-up the cropped surfaces of the bar, as well as allowing the workpiece to be located easily in the subsequent dies. The next operation, stage 2, is an upsetting process with the ends of the bar contained within conical dies.The billet spreads to fill these dies and also increases in thickness in the central region. The third stage is similar to the previous one, but in this operation the ends of the workpiece are also extruded into a narrower cavity, causing an area reduction of 28%. In the fourth and final stage, the ends of the workpiece are held in a container, and the central portion is upset into a die cavity to form the required shape of the disc. A small-strain axi-symmetric (2-D) elastic-plastic formulation was used by AlSened et al. [7.19] for the FE analysis of this process. The forging was assumed to be done slowly at room temperature on 0.1% C steel. The beta-stiffness Fig. 7.49 Stages in the production of an automobile spigot.
!
J ^ \ \ \ V
Sx\x\xwx\v\
stage 1
160
Practical applications
Fig. 7.50 Distorted reference grids predicted by finite-element analysis for the four stages in forging of spigot (super-grid technique). (a)
stage 1
(b)
(c)
(d)
7.5 Multi-stage processes
161
technique was used to model the frictional conditions at the interfaces between the workpiece and the dies. A value of friction factor m = 0.07 was selected to generate a frictional restraint similar to the phosphate and soap lubricant used in the experimental trials. Due to the severe deformation produced in this process, the FE mesh needed to be re-formed at intervals during the analysis. Eight-noded iso-parametric quadrilateral elements were used throughout. Figure 7.50 shows how an initiallyrectangular grid inscribed on the cross-section of the bar would have appeared at stages throughout the process. These results were obtained by referring information about FE nodal displacement back to the reference grid (super-grid technique). Throughout the analysis, the length of of the workpiece was reduced by 1% of its original value during each increment. Stage 1 was performed with a mesh of 125 elements representing one quarter of the original bar cross-section. This stage ended when the required degree of chamfering had been obtained on the ends. No re-meshing was required during this stage and the same mesh was carried over into the FE analysis of the second stage of the process. This time, the ends of the mesh were contained within 15° conical dies. This stage required 24 increments. The third stage started with a reformed regular mesh of 95 elements constructed within the profile obtained from the previous stage. Values of the important state variables were also obtained from the previous analysis and interpolated to the new nodal points. The third stage was carried out in 36 increments, with another re-meshing after 30 increments. The recently-reformed mesh from stage 3 was then used to start stage 4. The deformation was complete after another 30 increments. The whole analysis required 180 seconds of CPU time on a Honeywell DPS8 mainframe computer. The distorted grids in figure 7.50 clearly show the intense band of deformation in the central part of the disc of the spigot at the end of the process. In figure 7.51, experimental and FE values of VPN hardness are compared at the end of each of the four stages of the process. The FE predictions agree very well with the experimental results at the end of stages 1 and 3, but the agreement is not so close at the end of the other two stages. However, this may be due to variations in the material used in the experiments, since the hardness measurements were carried out on different production components, sectioned at different stages of the deformation. Comparison of figures 7.51c and d show that there appears to be a significant increase in the hardness in the shank of the experimental specimens during stage 4. Since the shank is held within a closely fitting container during this part of the process, there ought to be very little deformation, and very little change in strain distribution as a result. It should be noted that the FE program correctly predicted very similar distributions of hardness in the shank at the ends of stages 3 and 4.
Practical applications
162
Figure 7.52 shows the FE predictions of forming force for each of the stages as a function of machine stroke. Also shown is the total force for a 4-stage heading machine. Information of this type is obviously very useful in optimising the use of forging machinery. Fig. 7.51 Comparison of finite-element predicted and experimentally measured distributions of VPN hardness on cross-section of spigot during forging: (a) end of stage 1; (b) end of stage 2; (c) end of stage 3; (d) end of stage 4.
Expt
(d)
y
261
\ 255—*~ ^ \
260 X
250
/ /
VJJ 260
E)CDt
Expt
FE
275
^ ^
, 220^—
9 ?
"^*^^T" "230-^^ 230 O ^24 -
'215
\
^190 1O /
/ I
1 QQ
l^ O y
I/// J214 / /2\0
FE
\ P
7.5 Multi-stage processes
163
stage 1
1
1
1
1000 900 800
-
700
i
600 g 8
-
^
/
;
500
|
400 Total accumulated ——• force //
/
/
/
300 /
200
/
- / S
\ S^ \
1 2
y^ 1
1
100 I
1
3 4 5 6 7 Machine stroke mm
l
8
1
0 9
Fig. 7.52 Variation in forging force predicted by finite-element analysis for the four stages of spigot forging and total force in a four-stage heading machine.
7.5.2
Short hollow tube (gudgeon pin)
A four-stage forging process is typical for the production of 'gudgeon pins' (small lengths of tube). The first stage is cropping, and is not included here. The three stages modelled are as follows: Fig. 7.53 Stages in the forming of a gudgeon pin: upsetting (a), indenting (b) and backward extrusion (c). Initial dimensions of billet and container: billet diameter Do = 22.8 mm; billet height HO = 34.2 mm; container diameter Dc = 28.575 mm. Indentation with punch of diameter Dp = 19.05 mm. (b)
(a)
(c)
Dn
y*
* ^*s-
- y
y
s
164
Practical applications Stage 1: Simple upsetting in a closed container (figure 7.53a). The billet length/ diameter ratio was 1.5 after cropping. The die was filled after a reduction in height of the billet by 37%, reducing the length/diameter ratio to just below unity. Stage 2: Indentation of the dumped billet by 1 mm to guide the punch in the backward extrusion operation (figure 7.53b). Stage 3: Backward extrusion of the indented slug, with an area reduction of 1-8:1 (figure 7.53c). The finishing operation involves punching out the base of the cup. This controlled-fracture process is not modelled here.
The model material used for the analysis carried out by Al-Sened et al. [7.20] was commercially-pure aluminium. The experimental processes were lubricated with lanolin, m = 0.1, and friction was incorporated in the finite-element analysis using the beta-stiffness technique. An axi-symmetric elastic-plastic, small displacement, isothermal FE program was used. Eight-node iso-parametric quadrilateral elements were used and the initial FE mesh is shown in figure 7.54. Extensive re-meshing was necessary due to the extreme non-linearities associated with the corners of the backward extrusion punch for stage 3. No remeshing was required for the dumping and indentation phases (figures 7.55a-c) but the backward extrusion sequence (figures 7.55d-g) required a re-meshing operation every 5% penetration). This was an extremely laborious process at the time the analysis was conducted as no Fig. 7.54 Initial FE mesh for forming of a gudgeon pin.
7.5
Multi-stage processes
Fig. 7.55 Predicted grid distortions for the dumping, indentation and backward extrusion stages.
(b)
(a)
——
(c)
_ ——-
(d)
(f)
(e)
(g)
165
166
Practical applications
automatic re-meshing programs were available. For practical use of FE simulations of large deformations re-meshing programs are essential. The internal flow during the dumping operation could not be determined experimentally but the changes in the external geometry of the billets was consistent with observations. A double-barrelled shaped billet was produced by the deformation prior to the billet contacting the walls of the die - this was due to the height to diameter ratio of the initial billet. When the billet did contact the die wall, at the top and bottom, these parts were retarded due to die friction. The last part of the billet to contact the die was in the 'valley' between the ends of the billet. This is reflected in the predicted strain distribution figure 7.56, where the maximum strain (0.57) is seen to occur at the outer radius of the billet a little above the centre line. The predicted grid distortion obtained from the super-grid technique was in excellent agreement with experimentally observed distorted grids, figure 7.57. The 'rigid' zone with a curved high-deformation boundary, the flow of material around the punch corner, and the spread of the deformation zone throughout the base of the can during the post steady-state phase are all clearly visible in both the simulation and the experimental trials. Fig. 7.56 Generalised strain distribution at the end of the dumping stage.
0.48
7.5 Multi-stage processes
167
Fig. 7.57 Distorted grids from experimental studies of the backward extrusion stage.
References [7.1]
Pillinger, I., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. An elasticplastic three-dimensional finite-element analysis of the upsetting of rectangular blocks and experimental comparison. Int. J. Machine Tool Des. Res. 25, 229-43 (1985). [7.2] Nagamatsu, A. andTakuma, M. Experimental study of pressure and deformation of rectangular blocks in compression. /. Jap. Soc. for Techn. of Plasticity 14, no. 144, 49-57 (in Japanese) (1973). [7.3] Eames, A.J., Dean,T.A., Hartley, P. and Sturgess, C.E.N. An integrated computer system for forging die design and flow simulation. Proc. Int. Conf on Computer-Aided Production Engineering, ed. J.A. McGeough, Mechanical Engineering Pubs., pp. 231-6 (1986). [7.4] Hussin, A.A.M., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Simulation of industrial cold forming processes. Comm. App. Num. Methods 3, 415-26 (1987). [7.5] Clift, S.E., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Fracture initiation in plane-strain forging. Proc. 25th Int. Machine Tool Des. Res. Conf, ed. S.A. Tobias, Macmillan, pp. 413-19 (1985). [7.6] Clift, S.E. Identification of defect locations in forged products using the
168
[7.7]
[7.8] [7.9] [7.10]
[7.11]
[7.12] [7.13] [7.14] [7.15] [7.16] [7.17] [7.18] [7.19] [7.20]
References finite-element method. Ph. D. thesis, University of Birmingham, UK (1986) (unpublished). Hartley, P., Clift, S.E., Salimi-Namin, J., Sturgess, C.E.N. and Pillinger, I. The prediction of ductile fracture initiation in metalforming using a finite element model and various fracture criteria. Special issue of Res. Mechanica, 28, 269-93 (1989). Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Prediction of deformation and homogeneity in rim-disc forging. /. Mech. Wkg. Techn. 4, 145-54 (1980). Hussin, A.A.M., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Elasticplastic finite-element modelling of a cold-extrusion process using a microcomputer-based system. /. Mech. Wkg. Techn. 16, 7-19 (1988). Kato, K., Rowe, G.W., Sturgess, C.E.N., Hartley, P. and Pillinger, I. Fundamental deformation modes in open die forging - finite element analysis of open die forging I. /. Jap. Soc. for Techn. of Plasticity 277, no. 311, 1383-9 (in Japanese) (1986). Kato, K., Rowe, G.W., Sturgess, C.E.N., Hartley, P. and Pillinger, I. Classification of deformation modes and deformation property maps in open die forging - finite element analysis of open die forging II. /. Jap. Soc. for Techn. of Plasticity 277, no. 312, 67-74 (in Japanese) (1987). Pillinger, I., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. A three-dimensional finite-element analysis of the cold forging of a model aluminium connecting rod. Proc. Jnstn. Mech. Engrs. 199, no. C4, 319-24 (1985). Liu, C , Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Elastic-plastic finite-element modelling of cold rolling of strip. Int. J. Mech. Sci. 27, 531-41 (1985). Liu, C , Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Simulation of the cold rolling of strip using an elastic-plastic finite-element technique. Int. J. Mech. Sci. 27, 829-39 (1985). Hartley, P., Sturgess, C.E.N., Liu, C. and Rowe, G.W. Experimental and theoretical studies of workpiece deformation, stress and strain during flat rolling. Int. Mater. Rev. 34, 19-34 (1989). Lahoti, G.D., Akgerman, N., Oh, S.L and Altan, T. Computer-aided analysis of metal flow and stresses in plate rolling. /. Mech. Wkg. Techn. 4, 105-19 (1980). Liu, C , Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Finite-element modelling of deformation and spread in slab rolling. Int. J. Mech. Sci. 29, 271-83 (1987). Sparling, L.G.M. Formula for spread in hot flat rolling (quoting results based on unpublished work of R. Hill) Proc. Instn. Mech. Engrs. 175, 616^-40 (1961). Al-Sened, A.A.K., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Finiteelement analysis of a five-stage cold heading process. /. Mech. Wkg. Techn. 14, 225-34 (1987). Al-Sened, A.A.K., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Forming sequences in axi-symmetric cold-forging. Proc. 12th Nth. American Manufacturing Res. Conf, SME, pp. 151-8 (1984).
8
Future developments
8.1
INTRODUCTION
The programs described in this monograph have been developed specifically for application to metalforming, with the associated large plastic deformation and rotations produced by these processes. The emphasis throughout is on the flow of metal and the quality of the product in terms of homogeneity, residual stress and defect formation. Various other types of program are available commercially and in laboratories. The earliest ones were extensions of linear elastic problems into small plastic deformation. As such, they are useful in structural engineering design and failure analysis. They may however lead to serious errors in the large plastic deformation range. Others concentrate on plastic deformation to the exclusion of elastic, simulating the classical rigid-plastic material of slip-line field theory. The plastic flow may be represented as a form of viscous motion, either with a constant coefficient of viscosity or of a non-Newtonian type. These are probably most suited to modelling high-temperature deformation. It is then, of course, important to include a thermal analysis which may be independent or preferably fully coupled to the incremental plasticity analysis. Elastic stresses can be determined in such solutions by a subsequent elastic unloading analysis. It is also possible to combine elastic-plastic analysis of the workpiece with an elastic analysis of the tools. A survey of many of the programs commercially available in 1986 has been published [8.1]. Some very helpful direct comparisons are provided in that book. Taking a general view of the whole situation, it appears that there is likely to be a convergence of objectives if not of particular methods during the next decade. All formulations intended for use in metalforming are likely to include elastic and plastic deformation, with temperature, speed and strain-hardening influences.
170
Future developments
In this chapter we shall consider the main advances that still need to be made in modelling and software, followed by a brief survey of the main areas of application of the large-strain FE programs. The very rapid growth of the use of finite-element methods in many diverse areas of engineering, greatly aided by the enormous increase in power and reduction in cost of computers, suggests that these will soon be achieved and superseded by more ambitious objectives.
8.2
DEVELOPMENTS IN SOFTWARE
Highly complex design problems in structural, aeronautical, electronic and hydraulic engineering can now be solved by commercially-available linear finiteelement packages. The main area for development may be in improving the efficiency of programs handling very large matrices, for example by greater use of vectorisation. Pre- and post-processing techniques and the presentation of information have been advancing rapidly during the past few years. One of the problems now is the effective assimilation and use of the mass of information that can be generated. The problems of computer memory-space and operating time become more serious limitations in non-linear analysis. The iterative types of solution are clearly inefficient, but so far it has always proved necessary to use small increments for non-linear problems such as are inevitable in plasticity. Attention to second-order and perhaps third-order corrections has permitted larger increments without serious error, thus substantially reducing CPU time. All FE analysis requires a mesh to be generated at the outset. Automatic 2-D mesh generation has considerably reduced the manual labour of initial setting up, and improved the efficiency of the computation. Mesh generation in 3-D involves much greater problems. Some advance has been made with space filling and contour matching using tetrahedral elements but difficulties remain with the more conventional hexahedral or brick elements. The Delaunay method may be mentioned as a recent improvement [8.2]. In many metalforming analyses the overall deformation is large and the meshes may be very distorted locally. Remeshing is therefore required.
8.3
ADVANCES IN MODELLING
Apart from the need for automatic mesh generation, it is important to have a fuller understanding of the requirements of the mesh and the influence of the mesh itself on the final result. The effects of mesh size are obvious. It is desirable to have a fine enough mesh to reveal all the necessary detail of the deformation or other features, but this must be balanced against CPU time and costs. The aspect ratio and type of element can influence the result. As has been seen in
8.3 Advances in modelling
171
earlier chapters, simple triangular elements tend to be overstiff in large plastic deformation. Some intelligent routines for element selection would enhance the general utility of FE analysis packages. A book dealing with error estimation and mesh refinement has been published recently [8.3]. Modelling in 3-D produces particular difficulties, which are enhanced by the need to reduce the total number of elements involved. Boundary conditions also cause difficulties. It is necessary to match the initial geometric shape as well as possible, but also to introduce tests to determine when a node comes into contact with a rigid or elastic tool. This can be done geometrically, and the node then restored to the surface if it has apparently crossed the boundary. The problem is further complicated when nodes may leave a boundary surface. It is then necessary to determine whether the normal force has become tensile, before re-siting the node. The general problem of contact between two deformable bodies has been approached in a recent publication [8.4]. There is still little understanding of boundary frictional conditions. In civil engineering it is usual to assume some coefficient of friction relating the normal and tangential forces or stresses. For most metalforming operations this is not valid, since the tangential stress is limited to the shear yield stress, which is independent of the stress acting normal to the surface. It is convenient to assume that the frictional resistance is expressible as some fraction m of the yield stress k, such that 0
172
Future developments
The modelling of inhomogeneous, anisotropic and composite materials is still at a very early stage, but clearly has many practical applications. There is still doubt about the accuracy of calculation of hydrostatic stress in elastic-plastic problems, because of the large elastic stresses implied by small errors in volume calculation.
8.4
MATERIAL PROPERTIES
Until quite recently, the analytical techniques available for metalforming could deal only with highly idealised material properties. It was common to assume an isotropic, homogeneous, non-hardening rigid-plastic workpiece in slip-line field analysis. This produced surprisingly accurate general pictures of the deformation pattern and the forces, at least for 2-D problems. Since little use could be made of detailed material properties there was no incentive to measure more than fairly crude values of yield stress and, more recently for elastic analyses, the fracture toughness. FE programs can now, however, produce very detailed information about the distributions of strain, stress, strain-rate and temperature. Indeed, a major problem is to find experimental methods that are sufficiently detailed to check the analyses. To make full use of the available FE packages, it is highly desirable, if not essential, to obtain much more refined and accurate materials property data. Such data do not exist in the majority of instances, and they are very expensive to obtain. Even the test methods themselves are open to question. A simple upsetting test involves significant inhomogeneity of strain, apart from the dubious nature of the frictional contributions, so FE analyses of the tests are also required. The common hardness test, when subjected to FE analysis, is found to be highly complex, involving steep strain gradients. Its interpretation is by no means clear. The establishment of a reliable data bank is clearly desirable but equally clearly beyond the resources of a single university department or a single company. This would be a worthy task for a central European institute. The problem is exacerbated by the well-known variability of material within a standard specification. Very large differences in cutting tool life can, for example, be encountered even in the machining of free-cutting mild steel. A lack of data is even more apparent when composites are considered. These are of course of great importance not only in the reinforced polymer field but in aerospace technology with metal/ceramic materials. Extreme forms of composite are found in low-density materials with closed or open pores, which incidentally also pose very interesting questions for FE analysis. A collaborative study of materials properties and the FE analysis of composites may offer for the first time a proper understanding of the role of macroscopic
8.4
Material properties
173
inclusions of hard carbides on the one hand and soft manganese sulphide on the other. These two have been of great importance in metal cutting for many years.
8.5
POST-PROCESSING
The typical product of a finite-element analysis of metalforming is a matrix of strain distribution, possibly also with the local temperatures. The deviatoric and hydrostatic stress components are also available. It is then necessary to present these results in a form that can be readily assimilated by the investigator, or possibly used in a control system. The most easily understood form for metalforming is a distorted grid based upon a regular initial pattern. Since the distortions themselves produce errors in the FE analysis, it is not possible to work progressively through the whole deformation, up to 80% or more, using the original elements. The technique used is to re-mesh at suitable intervals, as has been described in Chapter 6. To retain the visual image, the original grid is stored in its distorted form, updated at each re-meshing, so that the final display corresponds to the distortion that would actually be seen on a gridded specimen [8.5]. These techniques are useful but the re-meshing in particular takes time, because of the need to average and transfer all the nodal and element centroid data. Improved re-meshing techniques would be valuable. For more detailed numerical consideration, it is useful to display the strain and other results in the form of contour maps. Improvements in these are no doubt possible, especially for 3-D examination. The FE results can be linked with conventional CAD graphics. Apart from the visual inspection of the results, from which very useful conclusions can be drawn, there is also a great potential in analysis of the numbers themselves. For example, a predetermined limit may be set to the allowable strain before reheating or inter-annealing the workpiece.The programs can then determine how many reheating operations are needed, and at what stage they should be inserted in the schedule. Taking this a little further, the shapes of preforms can be modified, perhaps eliminating one of the inter-anneals, with attendant cost saving. The possibility of cracks occurring can also be examined. At the present stage of knowledge, the onset of shear cracking can be predicted from the value of the total plastic work at a particular location. Again, the preform shapes may be modified to avoid this problem, but this type of analysis is still at an early stage and is capable of considerable refinement. It is not, for example, yet known precisely how tensile hydrostatic stress and shear stress interact. After process cracking has been avoided or overcome, there is still the very important aspect of service properties of the product. The fatigue life can, for
174
Future developments
example, be greatly influenced by the residual stresses from the working sequence. These depend very strongly on the strain and temperature history of the process, but so far the subject of residual stresses has received little attention in FE analysis.
8.6
EXPERT SYSTEMS
As we have seen, large-strain FEM provides a powerful tool for studying the deformation occurring in a forging operation. As such, it can be used to indicate how the forging operation can be improved, but it cannot design the forging dies, or determine the number of preform operations or choose the forging conditions. These tasks, traditionally the province of craftsmen with many years of experience, are increasingly being undertaken by 'expert systems' [8.6]. Expert systems are computer programs that emulate the cognitive, reasoning and explanatory processes of a human expert within a specialised field of interest, using theoretical or empirical rules and knowledge of the problem domain. The complementary natures of expert-system design and FE analysis make them eminently suitable for integration with a die design package. In such a package, the expert system would design the dies for each stage of the forging sequence, and specify the initial billet size and the forging conditions. For components that fall within established categories, this would be sufficient. An FE analysis would only be needed if it was uncertain whether the rules built in to the expert system could be applied satisfactorily to an unfamiliar component. The results of the FE analysis would help in the design of the dies in this instance, and they would also provide information about the general applicability of the expert-system rule base. The FE results might even suggest how the rules could be modified in order to deal with future components of a similar type. In this way, the range of application of the expert-system die-design procedure increases with time, while the need for expensive FE analysis becomes less and less. This integration of the die-design and FE programs requires the use of an Intelligent Knowledge-Based System (IKBS) to compare the components that are to be manufactured with previous examples and to decide whether FE analysis is required [8.7]. A start has already been made in achieving this kind of integrated facility though much work still needs to be done in this area.
8.7
HARDWARE
It is not the purpose of this monograph to discuss hardware in any detail. The rate of progress is anyway so rapid that such an account would be out of date by the time it was printed.
8.8
Applications in the future
175
T h e r e seem however to be two general trends as far as F E plasticity analysis is concerned. O n the o n e hand bigger and faster machines of very great cost, such as CRAY-2, are appearing. These clearly extend the capability, especially for 3-D analyses, and they will undoubtedly be used. O n the other hand, the advent of personal computers with 20-megabyte memories on hard disc and 0.5 megabytes or more internally has dramatically altered the situation. Whereas a few years ago even linear F E analysis was far beyond the capacity of microcomp u t e r s , it has now been demonstrated that plasticity problems can be solved. T h e accuracy is as good as that of a main-frame computer, and in terms of total turn-round time the 16-bit micros have the advantage of the immediate availability of the solution [8.8]. Their cost is a major factor allowing many organisations to u n d e r t a k e in-house F E analysis if they so decide. T h e r e will certainly be further advances, as evidenced already by the popularity of powerful workstations. These can be used to run the programs to examine 3-D problems, which require large amounts of storage, and so provide a real alternative to mainframe computing for complex and lengthy analyses.
8.8
APPLICATIONS IN THE FUTURE
A number of current applications have been discussed in Chapter 7. These will certainly be multiplied in the next few years, to include many of the variants on common processes. Here we shall briefly consider a few areas to which relatively little attention has yet been given in non-linear FE analysis. Possibly the largest is the whole field of polymer processing. Until reliable thermal analysis could be included there was little point in applying FE methods to these materials.There are severe problems, such as the very low conductivity and high temperature sensitivity of polymers. The properties also vary considerably according to structure and chain length, which themselves can be changed by mechanical processing. Strain rate is also an obviously important parameter, but unlike metals, the polymers may change significantly in density and properties by orientational crystallisation. Sheet forming has been aided by FE analysis, but much remains to be learnt about the analysis of anisotropic materials commonly encountered in the forming of rolled stock. The effects of anisotropy may also be important in extruded material. Composites have already been mentioned; these usually have a deliberately-produced severe anisotropy. One of the greater problems in primary metal and alloy forming is the production of homogeneous fine-grained material in a reliably reproducible way. FE analysis of the processes and of the materials themselves may be expected to advance the technology in this respect.
176
Future developments
Reproducibility of stock material is of increasing importance as automated or computer control of processes becomes more widespread. Although in principle it is possible to apply adaptive control in many processes, for much of the large-scale primary processing the individual time cycles do not permit large or even small changes from billet to billet. There may nevertheless be economic advantages and material savings through special processing or recovery of rogue billets detected by prior detailed non-destructive testing. This is of course only to be considered for the very high-cost materials such as are used in aerospace. Rigorous control of these products is clearly essential, including careful attention to residual stresses and the ensuing fatigue properties. In this context, it should be recognised that high-stress low-cycle fatigue is frequently as important as the more common high-cycle failure. FE analysis probably has a significant role to play in the developments of materials science. One obvious area is in the stress analysis associated with fracture toughness. Beyond this the actual process of tensile and shear cracking may be illuminated by fine-scale FE approaches. The simulation of friction has been mentioned as a problem in FE modelling. Quite apart from this, it is well known that the friction of metals and polymers itself is closely associated with the elastic, plastic and visco-elastic deformation properties. The stress fields around interacting asperities are little understood, and the interfacial conditions are described only in general terms. There is scope for more detailed analysis. The problems of wear are universal in machinery but scientifically wear is even less well understood. Wear rates are either measured empirically in each situation or some stochastic model is used. A detailed knowledge of the deformation and fracture processes at local contacts should advance the understanding of this commercially-important topic.
References [8.1] Niku-Lari, A. (Ed.) Structural Analysis Systems: Software, Hardware, Capability, Applications, Vol. 1, Pergamon (1986). [8.2] Cavendish, J.C., Field, D.A. and Frey, W.H. Automatic mesh generation: a finite element/computer aided geometric design interface. The Mathematics of Finite Elements and Applications V, Academic Press, pp. 83-96 (1985). [8.3] Babuska, I., Zienkiewicz, O.C., Gago, J. and de A. Oliveira, E.R. (Eds.) Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Wiley (1986). [8.4] Baaijens, F.P.T., Veldpaus, F.E. and Brekelmans, W.A.M. On the numerical simulation of contact problems in forming processes. Proc. 2nd Int. Conf on Numerical Methods in Industrial Forming Processes, ed. K.
References
[8.5] [8.6] [8.7] [8.8]
177
Mattiasson, A. Samuelsson, R.D. Wood and O.C. Zienkiewicz, Balkema Press, pp. 85-90 (1986). Al-Sened, A.A.K., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Forming sequences in axi-symmetric cold-forging. Proc. 12th Nth. American Manufacturing Res. Conf., SME, pp. 151-8 (1984). Vemuri, K.R., Raghupathi,P.S. andAltan,T. Automatic design of blocker forging dies. Proc. 14th Nth. American Manufacturing Res. Conf., SME, pp. 372-8 (1986). Rowe, G.W. An intelligent knowledge-based system to provide design and manufacturing data for forging. Computer-Aided Eng. /., 56-61 (Feb. 1987). Hussin, A.A.M., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Nonlinear finite-element analysis on microcomputers for metal forging. /. Strain Analysis 21, no. 4, 197-203 (1986).
Appendices Appendix 1 DERIVATION OF SMALL-STRAIN [B] MATRIX FOR 2-D TRIANGULAR ELEMENT With a linear interpolation function, the displacement (uh u2) of any point (*i, x2) within the element may be expressed as: ut = at + bnxx + bi2x2y i = 1,2
(Al.l)
(The bold subscripts that denote values for a particular element have been omitted for the sake of clarity. This will be the case for the rest of this appendix, though it is to be understood that all relationships apply to a given element of the FE mesh.) If the displacement of node / of the element is (dIh dI2) and this node has co-ordinates (xIh xI2) then: dn = 0i + bnxn + 612*12 d21 = ax + fcu*2i d31 = ax + bnx31 (with similar expressions for the displacement of nodes in the x2 direction.) Subtracting equation A1.2 from equation A1.3: d2l - dn =
bn(x21-xn)
Subtracting equation A1.2 from equation A1.4: d31 - du = 6n(*3i-* Multiplying equation Al.5 by (*3i-*n)/(*2i-*n) and subtracting from equation A1.6: (d31-dn)-(
(x22-x12)
\
or (d21-dn) (*32-*12) - (*3
Thus: bn =
+ (x2i-xn)dii (X2Z-X12)
1 Derivation of small-strain [B] matrix
179
But, with reference to figure Al.l, the area A of the triangle is given by: A
= ^1453 + ^3562_^1462
= — Ol2+*32) (AT 3i—ATn) + —(X32+X22) (X 2l-X3l)-
( )
( )
— (
(
So from equation A1.8: xn)d3i \
6
(AUO)
An analysis similar to the above gives: foil = — I
(x22-X32)dn +
+
(Al.ll)
*21 = — I
hl2=
TAX
di= — I
(* 11*22-* 12*20^3/
Now: dx1 ^22 =
bn
du2
Fig. Al.l Calculation of area of triangular element.
4
5
(A1.15) (A1.16)
180
Appendices du2
(A1.17)
so: € =
/dn\ *22-*32
0
-^32-^12
0
-^12-^22
0
*31"*21
0
^11-^31
0
^31-^21
-^22-^32
^11-^31
da d22 d3i
VW or € = [B]d
(A1.18)
Appendix 2
DERIVATION OF ELASTIC [D] MATRIX In Chapter 2 (equation 2.16) it was shown that: /
e n
-v
l
\
—v
0
0
0
-v
0
0
0
a22 (733
—*>
1
—v
-i,
1
0
0
0
0
0
0
2(1 + 1,)
0
0
723
0
0
0
0
2(1 + 1,)
0
Via/
0
0
0
0
o
:2(l + i,)_
e 22 e 33
1
?12
%
I O23
(A2.1) Clearly: (A2.2)
an =
so all that remains is to express t h e three normal c o m p o n e n t s of stress in terms of t h e strain c o m p o n e n t s . Multiplying e22 by v a n d adding to e n : E(en
+ ^622) = (1—^ 2)cr11 - 1^(1 + ^)0-33
(A2.3)
- 633) = ( 1 + I,)(7JI - ( 1 + I,)O33
(A2.4)
Subtracting e 33 from en: E(en
Multiplying equation A 2 . 4 by v and subtracting from equation A 2 . 3 : E[(l—v)eii
+ ^622 + ^€33]— (1 ~ v — v — v}(J\\ (A2.5)
so: (A2.6) and similarly: (722
=
+ (1-
+
(A2.7)
182
Appendices
<*33 -
(A2.8)
+ ve22
"
(i+*/)(i-:7vY
Hence: 1-V
V
V
1-2 v
l-2i/
l-2i/
V
1-v
V
\-2v
l-2i/
l-2i/
V
V
l-2i/
1-2 v
l-2i/
al2
0
0
0
O-23
0
a22 0-33
622
0 1
^33
0
T 0
0
0
T
0
0
723
—
713.
(A2.9) or:
(A2.10)
It is useful to be able to express the above relationship using subscript notation. Since y/;- = 2e,y: a-ij = 2Geih i ± j
(A2.ll)
where G is the Rigidity Modulus.For the normal components of stress, for example i = j = 1: 2G 2G
(A2.12)
= 2G(6U+— ^ using the usual suffix summation convention. Similar expressions may be obtained for the other two normal components of stress. Equations A2.11 and A2.12 may be combined using the Kronecker delta: <>rf +x ( v \ Oiy = 2G(6fy + By I — — I € Alternatively, we can write:
^ kk)
cr'ij = 2Ge'ij a n d o» = Kekk
(A2 13)
'
(A2.14)
2 Derivation of elastic [D] matrix
183
where the deviatoric components are defined by: h
o-tj = (Tij-dijcf1 and e'ij =
1
(A2.15)
€ir—Sij€ kk
6M is the sum of the normal components of strain (the bulk or volume strain), K is the Bulk Modulus and o^ is the hydrostatic stress, which is equal to one third of the sum of the three normal components of stress.
Appendix 3
DERIVATION OF ELASTIC-PLASTIC [D] MATRIX The basic assumption of the elastic-plastic theory is that an increment of strain can be separated into an elastic part, which may be recovered on unloading, and a plastic part, which results in permanent deformation. Since, for all practical purposes, the permanent deformation does not involve any change in volume, the bulk or volume strain occurring during a process must be recoverable and hence elastic. Thus, by equation A2.14, it is simply proportional to the change in the hydrostatic component of stress. It is only necessary, therefore, to consider the deviatoric components of the strain increment: de;,- = deVj + de?;
(A3.1)
where it is to be noted that defy = defy, since there is no volume component of plastic strain. Using the incremental form of equation A2.14, we can write immediately: d€
^ " = 2G
da lj
'
(A3-2)
With the von Mises yield criterion, the principle of the normality of the plastic strain increment to the yield locus [A3.1] leads to the Levy-Mises flow rule: = a'ijAX
( A 3 3)
dA is a proportionality factor. As will be shown later, this depends upon the elastic and plastic moduli, the current state of stress and the components of the strain increment. Substituting from equations A3.2 and A3.3 into A3.1 gives the Prandtl-Reuss elastic-plastic flow equations: de!/= — d ^ y + ^ y d A
(A3.4)
We require an expression giving the components of the stress increment in terms of the components of the strain increment and so must determine the proportionality factor dA. From equation A3.3 and the definition of an increment of generalised plastic strain given in Chapter 3:
3 Derivation of elastic-plastic [D] matrix
d6P =
185
,2
—C
=
\~3(Tii(rijJ
dA
dA = y
(A3.5)
a is the generalised stress which is also defined in Chapter 3. Define Y to be the rate of change of Y, the yield stress in simple tension, with respect to plastic strain. This quantity is obtained as required using the empirically-determined relationship between yield stress and plastic strain: Y'= — or Y'de p = dY (A3.6) dep But the von Mises yield criterion states that the yield stress equals the generalised stress, so: da dY=
doit (A3.7)
3
'
2a
'
Substituting from equations A3.5 and A3.7 into equation A3.6: 2 or: 4_ —r 2Yd\
= a'klda'kl
(A3.9)
But from equation A3.4: da'kl = 2G(de'kl - a'kldk)
(A3.10)
Hence: 4
"
"d\=a'kr2G(de'kl-a'kldk) 2 l-^-d dX
J
(A3.ll)
186
Appendices
Rearranging this equation:
~9°lY
+
T G ' ^ ) d A = G '°i / d €*'
(A3>12)
or: dA=-
-
a kld€ kl
'
' 5
(A3.13)
where:
4(^) Substituting equation A3.11 into equation A3.4 and rearranging:
(A3 15)
-
By definition of the deviatoric components of strain and stress (equation A2.15): (A3.16) or:
^"^"3
5—J
Thus the required constitutive relationship between the components of incremental strain and the components of incremental stress is:
*, =2G [ieil+Sll{^)iem-^pL\
(A3.18)
since oi/de£/ = o-'kidekh In matrix form: da = ([De] - [Z)P])d€
(A3.19)
3
Derivation of elastic-plastic [D] matrix
187
where the elastic matrix is: i-v
V
l-2i/
l-2i/
l-2i/
V
1-j/
*/
l-2i;
l-2i/
1-21/ l-i/
1-2
[De] = 2G
J/
1-2*
0
1-2*
1
0
0
0
0
0
0
0
y
0
0
0
0
y
0
0
0
0
0
1
(A3.20)
0 1 ~2
the plastic matrix is:
(A3.21)
^"23^22
^"23^33
^23^12
^23^23
and the stress and strain increment vectors are: do- = ( d o n , do-22, do-33, do-12, do-23, dcr 13 ) T
(A3.22)
d e = ( d e n , de 2 2 , d6 33 , dy 12? dy 2 3 , dy 1 3 ) T
(A3.23)
References [A3.1] Drucker, D.C. A more fundamental approach to plastic stress-strain relations. Proc. 1st Nat. Cong. App. Mech., ed. E. Sternberg, ASME, pp. 487-91 (1951).
Appendix 4
DERIVATION OF SMALL-STRAIN STIFFNESS MATRIX [K] FOR PLANE-STRESS TRIANGULAR ELEMENT To avoid the proliferation of subscripts, assume, without loss of generality, that the body consists of just one element. The element stiffness matrix is then the same as the global stiffness matrix [K] and is defined by: f = [K]d
(A4.1)
in which the global vectors of nodal force f and nodal displacement d are identical to the corresponding element vectors. The element is initially unstressed, with zero nodal forces, and the set of forces f is then applied to the nodes. A possible final configuration will be one in which the nodes have been displaced by amounts d. The potential energy of this new configuration (relative to the potential energy of the initial configuration) is the difference between the extra strain energy U stored in the element, and the work Wdone by the applied forces in moving their points of application between the initial and the final configurations. For plane-stress conditions: I = U-W
1f
= yj -fndn -fndu -f2\d2X -f22d22 -f31d31 -f32d32
(A4.2)
in which fn denotes the component of force in the xt direction acting t node /, and dn is the displacement of node / in the xt direction. For constant-strain triangular elements, the integral may be evaluated directly, and equation 2.24 may be used to express the stress components in terms of the strain components:
_v(
I — -~-\ (Dn€u -fndn
or:
\ + ^12^22)^11 + (^21^11 + D22e22)e22 + (^33712)7121
-fndu
-/2W21 ~/22^22 -/3W31 ~/32^32
VI
D2i)ene22
12 -f2id2i
(A4.3)
+ 2)33(712)
-f22d22 -f31d3l
-f32d32
(A4.4)
4 Derivation of small-strain stiffness matrix [K]
189
where D^ is the entry in row /, column j of the [D] matrix in equation 2.24. The actual configuration that the element takes up will be the one that minimises the potential energy of the system, i.e. the actual set of dn will be such that: dl ~dd~n
(A4.5)
= 0,7 =1,3 and/=1,2
For example, if I = 1 and / = 1: (Dl2+D2l)e22-
-/n = 0
-r^ da
(A4.6)
since e22 does not depend upon d n . Substituting for the strain components from equation A 1.18, and peforming the differentiation gives the result: 2(2^) 2 2Dl2 3
(x3l-x21) [ (x31-x21)dn +
22 +
-/u = 0
(A4.7)
in which use has been made of the fact that Di2 = D2h Rewriting equation A4.7 in vector form: T
/n
(
\(
\L>n \X22-X32)\X22-X32)
\ . r» /
A
H- Z>'33 (vX3i-X2ij(X3i-X2ij\
D12(X22-X32)(X31-X21)
+D
Dn
+ D33
(x22-x32)(x32-x12)
\(
I /
A
dl2 (x31-x21)(xn-x3l)
(2/1)^
d21 d22
Dn (x22-x32)(xi2-x22) \Dl2
(x22-x32)(x2i-xu)
+ D33
^
an
(x3l-x21)(x2i-xn)
d3l
+ 7) 33 (x31-x21) {xl2-x22)J
\ d32
(A4.8)
in which the transposed vector is the first row of the required element stiffness
190
Appendices
matrix [K\. The other rows may be found in a similar way by considering the other five expressions defined by equation A4.5. In matrix terms:
= yJo :redV-d f T
= yd T (j[Z?] T [D][B]dv)d-d T f
(A4.9)
Interpreting equation A4.5 as a vector expression, and differentiating equation A4.9 with respect to the vector d: 0 = J [B]T[D][B]dV-d - f
(A4.10)
so that: [K] = I [B]T[D][B]dV
(A4.ll)
For constant-strain triangular elements deformed under plane-stress conditions, it can readily be seen that equation A4.ll leads to the expression given in equation A4.8, and all the similar expressions for the other components of the force vector. However, equation A4.ll is quite generally true for all smallstrain formulations, using the appropriate definitions for [B] and [D].
Appendix 5
SOLUTION OF STIFFNESS EQUATIONS BY GAUSSIAN ELIMINATION AND BACKSUBSTITUTION The assembled global stiffness relationship: = f
(A5.1)
must be solved for the displacement vector d, given the applied force vector f. This relationship consists of a set of n linear simultaneous equations in n unknowns, where n is the number of degrees of freedom in the FE discretisation. Thus equation A5.1 may be written as: Kn
Kyi
K21
K22
Kl
A
A d2 \
(A5.2)
= Ka
K
n2
fa
w
The solution of this set of equations by Gaussian elimination and back-substitution is a two-stage process. The first stage, the elimination, converts the set of equations into an equivalent ordered set in which each equation involves one less variable than the preceding one. To do this, use is made of the fact that the solution of a set of equations is unchanged if a multiple of one equation is added, coefficient by coefficient, to another, as long as the same multiple of the right-hand side of this equation is added to the right-hand side of the other. We do not prohibit adding a multiple of one equation to the same equation, that is multiplying this equation by some number, providing this number is not zero. The proof of the above result is
192
Appendices
straightforward, for if dp, (/3 = l,n) is a solution for the set of equations, it is certainly a solution for equations a and y: =/«
(A5.3)
0=1
(A5.4) j3=l
It w times equation a is now added to equation y:
+ 0=1
3=1
3=1
=fy+wfa
(A5.5)
so the set of dp is still a solution of the new equation y. Returning to equation A5.2, a multiple of w = K21/Kn times the first equation can be subtracted from the second: K\n K2i-wKn
K22-wKu
K2n-wKln
Ka\
K,al
Kan
^nl
Kn2
Knn
(A5.6)
or:
fa
Ww d
^22
•
K*2n
f '\
1 d2 \ (A5.7)
Ka\
^al
K
n2
Kan
Knn
fa
\J w
5 Solution of stiffness equations
193
where the asterisks denote that the quantities no longer have their initial values. Continuing this process, by subtracting a multiple of Kal/Kn times the first equation from each equation a, a = 3,n gives:
K12
K\n
(A5.8)
WW This process is then repeated, subtracting appropriate multiples of the second equation from all succeeding equations to introduce zeros into the second column below the diagonal, and so on. In general, this can be codified into the following algorithm: repeat for /3 = l,/i— repeat for a = repeat for y = subtract Kl from K*ay end repeat subtract K*a(j%IK%p from /* end repeat end repeat Note that in the inner loops, the repeated subscript j8 does not imply a summation. It should also be noted that the above piece of pseudo-code is several removes from an actual computer program. For instance, in practice the computationally time-consuming division would be removed from the innermost loop and instead a multiplication by a previously-calculated inverse would be used. With this algorithm, no entry in the matrix is actually set to zero, it is just ignored in all future calculations.
194
Appendices
At the end of the elimination stage, the equations look like:
n (A5.9) 0
0
0
0
da
w
0
w
where asterisks have been added to the first row of coefficients for consistency, even though these values have not been altered. The second or back-substitution stage of the solution procedure calculates the values of the components of the solution vector d by considering, in reverse order, the equations obtained in the first part: repeat for a = n,l displacementd a=lflB=a+1
end repeat in which the summation is taken to be zero if a = n. It can be seen that when the time comes to calculate any particular da, the dp values that occur on the right-hand side of the expression in the above loop are already known. In theory, the process of Gaussian elimination and back-substitution will solve any set of simultaneous equations providing the matrix of coefficients is non-singular, that is, has an inverse. (If the matrix is singular, a zero diagonal term will be encountered during the elimination phase, and an attempt will be made to divide by zero. This is unlikely to happen if the stiffness equations are based upon proper physical principles.) In practice, a computer implementation of the above solution procedure will be subject to accumulated round-off error, particularly as the method involves repeated subtraction of one quantity from another and these quantities may quite easily have approximately the same value. (Round-off error is a consequence of the fact that computers can only represent real numbers to a finite number of significant figures. The effective number of significant figures is decreased even further when a number is subtracted from one of about the same value because the higher-order figures cancel.) In the worst cases, when the matrix equations are ill-conditioned (roughly
5
Solution of stiffness equations
195
speaking, when the matrix is almost singular) accumulated round-off error may cause very small or zero diagonal coefficients to be calculated, leading to floatingpoint overflow, or division by zero. F E stiffness equations are usually well-conditioned, but even so the effect of round-off error will be to reduce the accuracy of the calculated solution of the equations. T h e standard way to reduce the effects of found-off error is to re-order a calculation so as to avoid, as far as possible, the subtraction of quantities of a similar size. In the Gaussian elimination procedure, this may be done by re-ordering the equations, each time a column of zeros is to be introduced, in order to place the largest coefficient in that column on the diagonal. This is called partial pivotting. Full pivotting, in which the unknowns of the equations are also reordered, may be carried out, but at the expense of having to keep track of the new orderings. T h e solution procedure outlined in this appendix has m a d e no special assumptions about the matrix [K], except that it must be non-singular. For isotropic flow however, [K] will always be symmetric. Examination of the elimination algorithm stated above shows that the symmetry of the submatrix to the right and below the current diagonal coefficient is preserved. This has two consequences. Firstly, only about half of [K] need be assembled and stored, and secondly only those calculations affecting the coefficients of the stored half of the matrix need be performed. Modification of the elimination algorithm to take advantage of these two facts is straightforward and is left to the reader. Modifications of the basic technique can also be m a d e to take advantage of the sparse (banded) nature of the stiffness matrix.This is discussed in Chapter 6.
Appendix 6
IMPOSITION OF BOUNDARY CONDITIONS A6.1
Components of displacement prescribed in global axis system
The basic Gaussian elimination and back-substitution procedure solves the stiffness equations to obtain a nodal displacement vector d, given an applied nodal force vector f. In metalforming problems, the effective force acting at all nodes is zero, except for those nodes presumed to be in contact with the dies, or for those nodes otherwise constrained - on a plane of symmetry for example. In general, the value of the force acting at such a boundary node is not known beforehand, though the value of at least some of the components of its displacement will be. Any solution procedure adopted in an FE metalforming program must therefore be capable of imposing prescribed values of certain components of displacement upon the solution, and calculating, as a result, the components of the reactive force. In the simplest case, one or more of the global Cartesian components of displacement at a node may be prescribed by the boundary conditions of the FE model of the metalforming process. A prescribed value may, of course, be zero. Suppose component da of the global nodal displacement vector is known. Consider the stiffness matrix equations at the stage in the Gaussian elimination procedure when zeros are to be introduced into column a below the diagonal (Appendix 5): K 11 i
22
K 2a
/h\\
« .
/ d2
« .
da
//A /I \ (A6.1)
=!
0
0
W
6 Imposition of boundary conditions
197
Since the value of the reaction fa is, as yet, undetermined, so is the modified value /*. Thus it is not possible to subtract multiples of equation a from all the succeeding equations as would normally be the case. However, since da is known, each of the coefficients in column a below the diagonal can be replaced by zero, providing da times the coefficient is subtracted from each of the right-hand sides:
n n
Kl
K*
0
0
n
K*
0
W ww
(A6.2)
The Gaussian elimination algorithm given in Appendix 5 may therefore be modified: repeat for a = l,n if da prescribed then
/« = o
end if end repeat repeat for /3 = l,n-l repeat for a = j8+l,n if dp prescribed then subtract Kip dp from /* otherwise repeat for y = fi+l,n subtract K*apK%yIK*pp from K*ay
end repeat subtract K*apf%IK*pp from /* end if end repeat end repeat and similarly, the back-substitution algorithm becomes:
Appendices
198
repeat for a = n,\ if da prescribed then reaction/^ = 2 ,
K
tpdp-ft
otherwise displacement da = \f*a-
2*
K
Udp)/K™
end if end repeat As in the previous appendix, a summation from n +1 to n is assumed to be zero.
A6.2
Components of displacement prescribed in rotated axis system
In general, die surfaces and planes of symmetry may not be aligned with the global axes. If the displacement of node / has some prescribed value in a given arbitrary direction, but unknown values in directions perpendicular to this, then all the global components of displacement will be initially unknown, as will all the global components of force. The method cannot therefore be used. The same thing applies if the displacement is prescribed in two orthogonal directions and unknown in a third. (If the displacement is prescribed in three orthogonal directions, the components in the global axis system can be found immediately, and the previous technique can be used.) Consider, as before the stiffness equations during the elimination stage, but this time at the point when zeros are to be introduced below the diagonal into the column corresponding to the first degree of freedom at node /. Rewrite the matrix equations in terms of NxN nodal submatrices [KJJ] containing the four (2-D) or nine (3-D) coefficients relating f7, the global components of force at node / to dy, the global components of displacement at node / : [K*lN]
*!,] [K*n] [K h] [0]
[K*2N]
h
1n \
d2
[K*22]
(A6.3)
= [K*IN] [0]
[0] [K*NI]
[0]
d,
[0]
[K*NN]
W
6 Imposition of boundary conditions
199
in which, as before, asterisks denote that the quantities no longer have their original values. It should be noted that the diagonal submatrices above row / are themselves in upper-triangular form. Define a new set of orthogonal axes Xt such that one of these axes is parallel to the prescribed displacement of node / (or two of the axes are so aligned if the displacement is prescribed in two perpendicular directions). Figure A6.1 shows the situation in 3-D with one prescribed component. Let [R] be the rotational transformation matrix containing the direction cosines of the three new axes with respect to the global Cartesian axes xt\ dXi
(A6.4)
where / denotes one of the rows of [/?], and j one of the columns. Denote the components of displacement of node / in the rotated axis system by d}. By definition, [R] is an orthonormal matrix. The expressions for the change of basis (equation A8.21, Appendix 8) then give that: (A6.5)
= [R]di
and similarly, if the components of f* are f*': f* = [R]t*>
(A6.6)
Fig. A6.1 Local rotated axis system at node. prescribed displacement of node - prescribed component node
Global Cartesian axes
^components 1\ to be evaluated Rotated Cartesian axes
200
Appendices
Substituting equations A6.5 and A6.6 into A6.3: [K*n]
[K*l2]
•
[K* u]
[0]
[Kh]
•
[K*2I]
•
[K*1N] [K*2N]
d2
(A6.7) [0]
[0]
[0]
[0]
•
[Kh]
•
[K*IN]
[K*NI]
•
[K*NN]
dN
VJ
which is equivalent to the two sets of equations di [K*1N] [0]
[Kh]
[K*22]
d2
[K*2N]
(A6.8a)
[0]
[0]
and:
[0]
[0]
[R]T[K*,N]
(A6.8b)
[0]
[0]
•
[K%,][R]
•
[K*NN]
6 Imposition of boundary conditions
201
since the inverse of an orthonormal matrix is its transpose. The Gaussian elimination may now proceed as before, with zeros being introduced into the two or three columns of the second matrix corresponding to the components of displacement of node /. This may be accomplished either by subtracting multiples of one of the two or three equations expressing the force at node /, or by modification of the right-hand sides, according to whether the particular component of displacement at node / in the rotated axis system is an unknown or is prescribed. Back-substitution is also carried out as described earlier except that when the rotated components of displacement and force have been evaluated at node /, they must be pre-multiplied by [R] in order to obtain the components of these nodal vectors in the global axis system. The global components of displacement at node / a r e then used in the back-substitution process to obtain the displacement at nodes 1 to / - I . It should be noted that, providing the original stiffness matrix was symmetric, the part of the matrix to the right of and including the submatrix [i?]T[Z£//]|7?] in equation A6.8b is still symmetric, so only the two or three rows of this matrix corresponding to node / actually need to be modified (assuming the upper triangle of the matrix is being stored).
Appendix 7
RELATIONSHIP BETWEEN ELASTIC MODULI E, G AND K Rigidity Modulus G Consider a unit cube of material subjected to pure shear, with engineering shear strain y12 in the xxx2 plane (figure A7.1). Since there is no stress or strain in the x^ direction, this axis has been omitted from the diagram. As a result of the application of the complementary shear stresses cr12 and o-2i, the square crosssection ABCD is deformed into the rhombus AB'C'D'. The engineering shear strain and shear stress are related by the Rigidity or shear Modulus G:
r,= f
(A7.i)
By constructing Mohr's circles for this stress system, or by considering, for example, the equilibrium of forces acting upon triangle B'C'D', it can be shown that the normal stress acting in tension along the diagonal AC is:
= al2
(A7.2)
and similarly, the normal stress acting in compression along the diagonal B'D' is: crBD' = -an
(A7.3)
Fig. A7.1 Unit cube subjected to pure shear.
D
~T 1 \
A
"21
D'
N\ / / /K X T
77 2 ~" 2
C' {
B' B
7 Relationship between elastic moduli
203
The normal strain in the original diagonal BD, e BD , resulting from these two normal stresses (cf. equation 2.14) is therefore: eBD=—(cj
E
B D
' '-vaAC)
E ""
(A7.4)
Applying the cosine rule to the triangle AB'D'\ (B'D'f = {AB'f + {AD'f -2AB' AD' cos ( y - 7 1 2 ) = 2-2y 1 2
(A7.5)
for small angle y12. But: BD(1 + e BD ) = V2(l + e BD )
(A7.6)
2(1 + e BD ) 2 = 2(1 - yl2)
(A7.7)
B'D'= so:
and ignoring the second-order term involving the square of the strain: €BD = - y
(A7.8)
Thus, from equations A7.4 and A7.8:
eBD=- Y=-~ir(Ti2
(A79)
and comparing with equation A7.1: G =Bulk Modulus K The elastic Bulk Modulus determines the relationship between the bulk strain, the sum of the normal strain components eih and the hydrostatic stress ah = 07,73: ah = K€H
(A7.ll)
But as was shown in Chapter 2: 1
6n = — (o-n-^a-22-w 6
1
33 )
22 = ~pr ( - ^ c r n + cr22 - ^C733)
1 = —{-van ~
(A7.12)
204
Appendices
so summing these equations: en + 622 + 633 = — (l-2v)(a
n
+ a22 + 0-33)
(A7.13)
Hence: (1-21/) 3(l-2i/) h €// = — g — O}y = (Th
(A7.14)
and so: E 3(1-2./)
(A7.15)
Appendix 8
VECTORS AND TENSORS Let {v} be the set of all vectors in 3-D geometric space. This set forms a vector space over the scalar field of real numbers, in which vector addition and scalar multiplication have their usual geometric interpretations. It is therefore possible to choose a set of three linearly-independent vectors g, to form a basis of {v}. (A set of three vectors is linearly independent if none of them may be expressed as a linear combination of the other two - in geometric terms this means that the three vectors do not lie in the same plane.) Any member of {v} may then be written in the form: v = v'gt (A8.1) in which the scalar quantities vl are called the contravariant components of v with respect to the basis g, (figure A8.1). Equation A8.1 illustrates the summation convention. This states that whenever an index occurs exactly twice in a term of an expression, that term is summed over all three values of the repeated index. The repeated index is called the dummy index and may be chosen arbitrarily, except that it may not be the same as any other index occurring in the same term. In addition, if the expression is referred to a basis of {v}, one of each pair of dummy indices must be a subscript and the other must be a superscript. This last requirement is relaxed for the special case of an orthonormal basis (see below). Any index which occurs only once in a term is called a free index. The range Fig. A8.1 Contravariant components of a vector.
206
Appendices
convention states that an expression refers, in turn, to all three values of all the free indices. If indexed expressions are equated, the equality is taken to be true for the complete range of the free indices and so any free index must be present on both sides of the equality sign and all occurrences must either be as subscripts or as superscripts. Again, the last restriction does not apply when an orthonormal basis is being used. Define the scalar product (•) of two vectors to be: u v = KV(&-g;.)
(A8.2)
g/gy = g/g/ = g/Hg/|cos(a)
(A8.3)
for:
where | | represents the magnitude of the enclosed vector, and a is the angle between g, and g; in 3-D space. Let By be the Kronecker delta in which it is understood that either of the indices may be a subscript or a superscript. This quantity is defined to be equal to one if the values of the indices are the same, and zero otherwise. For the basis gh define the dual basis g/ such that: g/g ; = V
(A8.4)
Using the dual basis, an alternative expression for the vector v may be obtained: v = v/g<'
(A8.5)
in which vx are called the covariant components of v (figure A8.2). A basis is said to be orthonormal if it is equal to its dual. Under these circumstances, the contravariant and covariant components of a vector are the same, and by convention subscripts are used for all indices. Suppose a different basis Gy of {v} is chosen. The position vector x of an arbitrary point in 3-D space may be written as: x
Fig. A8.2 Dual basis. 83
= x
(A8.6)
8 Vectors and tensors
207
and: x = Xig = XjGj
(A8.7) k
Taking the scalar product of the vectors in equation A8.6 with g gives: xigk'gi = X^gk'Gj
(A8.8)
or, using equation A8.4: xk = Xjgk-Gj
(A8.9)
Similarly: xk = Xjgk'Gj
(A8.10)
Xk = x%-Gk
(A8.ll)
Xk = Xjg>'Gk (A8.12) Differentiating these expressions with respect to the three co-ordinates gives:
£-S-<-°'
-
and: dXi dX>
axr**'*'*'
(A8 14)
-
Substituting equation A8.9 into equation A8.6: XiGi = Xi{%LGfa
(A8.15)
or, using equation A8.13: X ' G , = X ' ^ . g,
(A8.16)
But the choice of x was arbitrary, so: G, = ^ - g,
(A8.17)
If V and Vj are the contravariant and covariant components of v with respect to the new basis Gy, then by analogy with equations A8.ll and A8.12: yi = v'gfG'
(A8.18)
Vj= Vig-Gj
(A8.19)
which, using equations A8.13 and A8.14 may be written as: Vj=vl —j dx
(A8.20)
V, =v, ^
(A8.21)
208
Appendices
These two equations lead to the alternative definition of a vector: a set of three scalar values obtained in co-ordinate system xl are the contravariant components of a vector if the corresponding values calculated in co-ordinate system X with the same origin may be expressed in the form of equation A8.20. If the values may be related by an expression of the form of equation A8.21, they are covariant components. For example, the three values defining the force acting at a point are contravariant components, whereas it can be easily shown that the three quantities obtained by differentiating the equation of a plane with respect to the three co-ordinates are the covariant components of a normal vector to that plane. Note that although the theory so far has only been concerned with geometric space, the above definition is quite general, since any 3-D vector (such as force) may be mapped onto an appropriate geometric vector. Vectors are a special case of tensors i.e. first-order tensors. Conversely, higherorder tensors may be expressed in terms of products or quotients of vector components. The definition of a tensor is simply an extension of that just stated for a vector. For example, the nine scalar values tij obtained in co-ordinate system xl are the contravariant components of a tensor if the corresponding values Tkl obtained in co-ordinate system Xk are related to these by: .. .. dXk dXl Tkl=tl] ax* dxj Similar definitions exist for covariant and mixed tensors. One important example of a tensor quantity is, of course, the stress in a deforming body.
Appendix 9
STRESS IN A DEFORMING BODY Consider an infinitesimally-small region of the body located at point P at time t. At some later time, t+dt, this region has deformed and is situated at point P' (figure A9.1). During this increment of deformation, an infinitesimally-small plane surface at P, with area da and unit normal n is deformed into a plane surface with area da' and unit normal n'. Let g/ be the basis of a stationary reference co-ordinate system. At time t+dt, choose a new co-ordinate origin O' and a new basis G, so that the co-ordinates of P' with respect to G,- are the same as the reference co-ordinates of P. The new basis and origin define the convected co-ordinate system. Uppercase letters will be used for convected components and lower-case letters will be used for reference components. A prime will be used to indicate that a particular quantity refers to the deformed state at time t+dt. Suppose the reference co-ordinates of P and P' are xl and x11 respectively, so that xn = xtl{xj,dt) is the function defining the deformation of the infinitesimal region originally situated at P. Then: O'P' = X% = x'% - OO'
(A9.1)
or, since XfJ are convected co-ordinates: j = xfigt - OO'
so that:
1 x" = jt'Gyg ' + OO'-g*"
Fig. A9.1 Deformation of infinitesimal region.
(A9.2) (A9.3)
Appendices
210
Differentiating and comparing with equation A8.13 (Appendix 8), noting that this expression is independent of the position of the origin of the two co-ordinate systems, results in the relationship: dxn or:
dxl
^A94)
dxl
(A9.5)
in which the superscript',/' denotes the derivative with respect to the yth reference co-ordinate. Another important result concerns the deformation of the infinitesimally-small plane originally situated at P (figure A9.2). Since n is a unit vector, the definition of vector product (*) gives that: nda = v W (A9.6) where v1 and v2 are infinitesimally-small vectors bounding area da at time t. At time r+dr, these are deformed into v' 1 and v'2, so: n'da' = v ' W 2
(A9.7)
3
Choose an arbitrary small vector v in the undeformed body and write: (A9.8) Thus: y"
= v'ijGj = vijGj
(A9.9)
by the property of convected co-ordinates, since V are infinitesimally-small vectors. Taking the scalar product of v' 3 with the deformed area normal having covariant components N'jda': (v' 3 -n')da'
=V'3iN'jdaf(GrGj) = v3iN'ida'
(A9.10)
Fig. A9.2 Undeformed and deformed area elements. n
n'
9 Stress in a deforming body
211
from the definition of dual basis given in equation A8.4 (Appendix 8). Alternatively: (\'3-ri)daf =(v' 1 *v' 2 )-v' 3 = (V'lkGk*V'2mGm)'V'3nGn = det(V/iy)(Gi*G2)-G3
(A9.ll)
by the definition of triple product, where det ( ) denotes the determinant of the enclosed matrix. Substitution of equations A8.17 (Appendix 8) and A9.5 then gives that:
= det(jc'I"'/)(v1*v2) • v3
(A9.12)
v3W;-dfl' = \'3-n'da'=J\*-nda =Jv3intda
(A9.13)
Thus:
or since the choice of v3 is arbitrary: N'ida' = Jriida
(A9.14)
in which / is the Jacobian of the deformation i.e. the determinant of the matrix of the transformation of the reference co-ordinates of particle P during the time interval dt (equivalent to the ratio of deformed to undeformed infinitesimal volumes). The stress acting at a point is a tensor describing the force per unit area acting upon any infinitesimal plane situated at that point. The reference components of True or Cauchy stress are denoted by o-iJ' and are defined by: /'7 = o-'^J da'
(A9.15)
fJ
Where f are the contravariant reference components of the force vector f' acting upon the deformed infinitesimal plane, which has area da' and a unit normal with covariant reference components n\. It can be easily shown from equation A9.15 that aij obey the co-ordinate transformation rule stated in equation A8.22 (Appendix 8), so these are contravariant components. The variational principle used in Chapter 5 is expressed in terms of the rate of nominal stress (also called the Lagrange or Piola-Kirchhoff I stress). The nominal stress sij is calculated assuming that the forces in the deformed configuration act on the body in its undeformed state. Thus: f'j = sijnida
(A9.16)
The rules for change of basis state that: n\ = N'k-^ dx
(A9.17)
212
Appendices
so that equation A9.15 may be written as: / ' ' =crlj—rN' kda'
= skinkda
(A9.18)
dX
Using equation A9.14, this gives the result: Jalj—
T
= skj
(A9.19)
dX
or:
JaV-sP £=&«.*
(A9.20)
The conditions for force equilibrium require Cauchy stress to be symmetric, but equation A9.20 shows that, in general, the same is not true for nominal stress. A third types of stress, the Kirchhoff or Piola-Kirchhoff II stress, may be defined in the following manner. Consider the undeformed infinitesimal surface with area da. This has a unit normal with covariant components nt in the reference system. As mentioned above, the covariant components of the normal to a plane in a co-ordinate system are obtained by differentiating the equation of that plane with respect to the three co-ordinates in turn. But by the definition of convected co-ordinates, the equation of the undeformed plane in the reference system is the same as the equation of the deformed plane in the convected system. Hence n{ are proportional to N't (though not necessarily equal). Define f to be the force acting upon a plane, in the deformed body, which has area da' and a normal with reference components N'j. That is: / ' = o-'WJda'
(A9.21)
ij
The Kirchhoff stress r is then defined so that this is the force which would be calculated to act upon the undeformed infinitesimal plane: / ' = Tij'ni da
(A9.22)
Using equation A9.14 Kirchhoff stress may be written in terms of Cauchy stress and nominal stress: TiJ
= Jo-ij = skjxfi'k
(A9.23)
The following example will illustrate the difference between the three definitions of stress. Figure A9.3 shows a cubic element of unit dimension. The orthonormal basis gz define a Cartesian reference co-ordinate system jt'.The element is first extended in simple tension by amount e in the xl direction (with Poisson contraction ve in the other two directions) and then rotated by an angle a about the x3 axis. The edges of the deformed element define the convected basis G,. Note that although G/ is an orthogonal basis, it is not orthonormal.
9 Stress in a deforming body
213
The deformation matrix for this example is: 0
x">J = -
dx[
~dXj
(l+e)sin(a)
(l-z^)cos(a)
0
0
0
l-ve l-ve
and so:
(A9.24)
= (l+e)(\-vef
(A9.25)
Suppose that at the end of this simple deformation there is a tensile stress of TN/m2 acting in the X1 direction. Equilibrium of force (figure A9.4) means that the reference components of Cauchy stress are: Tcos2(a) Tsin(o:)cos(a) 0 Tsin(a)cos(a)
Tsin2(a)
0
0
0
0
Fig. A9.3 Deformation of unit element.
x2
rotation
(A9.26)
214
Appendices
-TT.lLz^o,
^ = (1 + e)(cos(a)g1 + sin(a)g2) 81
Fig. A9.4 Cauchy stress. The nominal stress sij in this example (figure A9.5) is: T(l~ve)2cos(a)
T(l-^) 2 sin(a)
0
0
0
0
sij =
(A9.27)
The Kirchhoff stress Ti} (figure A9.6) is: r(l+e)(l-^) 2 cos 2 (a) (A9.28)
Fig. A9.5 Nominal stress.
n' = (l +e)G1
9 Stress in a deforming body
215
da' = (1 - ve)2
cos 2 (a)g 1
Fig. A9.6 Kirchhoff stress.
Thus if e = 0.01, a is 5° and v is 0.5 (for a plastically-deforming body):
=T
T'i
=J
0.992404
0.086824
0
0.086824
0.007596
0
0
0
0
" 0.986258
0.086286
0
0
0
0
0
0
0
0.992330
0.086818
0
0.086818
0.007596
0
0
0
0
(A9.29)
(A9.30)
(A9.31)
from which it can be seen that the Cauchy and Kirchhoff stress measures are, to a good approximation, the same for conditions of near incompressibility.
Appendix 10
STRESS RATES The FE governing equations are expressed in terms of stress rates. These are evaluated at the start of each increment (time t) when the reference and convected co-ordinate systems coincide. Therefore, at this time, all three stresses (Cauchy, nominal, Kirchhoff) have the same value. However, their time derivatives are, in general, not equal. The variational principle involves the rate of nominal stress. Therefore, differentiating equation A9.23 (Appendix 9) with respect to time gives: = P + crVu1* lJ
ij
(A10.1)
lyj
since x' = 8 at the start of the increment. u is the rate of deformation tensor, the time derivative of xrltj. In the constitutive relation, both the stress rate and the strain rate must be independent of any rigid-body component of the deformation. As explained in Chapter 5, Kirchhoff stress is the appropriate measure to use, and so the constitutive law is expressed in terms of the Jaumann rate of this stress.This is defined to be the rate of change of the components of Kirchhoff stress calculated in a co-ordinate system which rotates with the material. The deformation tensor is clearly real, and is non-singular since the inverse transformation may easily be defined. It may therefore be uniquely expressed as a product of a rotational transformation r ^ and a symmetric deformation q^: xn,j
=
rikqkj
(A10.2)
Differentiating this expression with respect to time: = fij + qij i}
(A10.3)
iJ
since r^ = q ' = 8 ' at the start of the increment. Therefore: 1 .. .. rl} = —(u l']-uhl) = J>
(A10.4)
where the angle brackets denote the skew-symmetric part of the enclosed matrix. Define a transformation which is the rotational part of x"'', that is: x*" = rij
(A10.5)
10
then the rotationally-invariant Kirchhoff stress, ij _
=
T**7
*mn_, *i,m
Stress rates
217
is given by:
*j,n
mn im jn T* r r
(A10.6)
using equation A8.22 (Appendix 8) for the transformation of the components of a tensor. Differentiating this expression with respect to time and noting again that rli = 8ij' at the start of the increment: j . ij _
j . * ij _^_ T * mjf im _j_ T * in^ jn
= r*ij + (ikj + aik
(A10.7)
from equation A10.4 and using the fact that rotationally-invariant Kirchhoff stress is the same as the reference components of Cauchy stress at the start of the increment. i*iJ' is the Jaumann rate of Kirchhoff stress. Combining equations A10.1 and A10.7: = T*ij-(Tkjelk-(jlkekj+(Tlkuj'k
(A10.8)
ij
where e , the strain-rate tensor is defined to be the symmetric part of the deformation-rate tensor: g« =.!(,{'•./ +,}>.<•)
(A10.9)
Finally, since the reference co-ordinate system is chosen to be the orthonormal Cartesian system, equation A10.8 may be rewritten as: Sij = T*y - o-kjeik - dikekj +
(A10.10)
Appendix 11
LISTING OF BASIC PROGRAM FOR SMALL-DEFORMATION ELASTICPLASTIC FE ANALYSIS The program listed in this appendix was designed principally for demonstration purposes and to form an introduction to non-linear FE plasticity analyses. In order to simplify the approach, only constant-strain triangular elements have been used. The program is sufficiently flexible, however, to be able to deal with plane-stress, plane-strain and axi-symmetric examples. The data contained within the sub-program FILDATA are specifically for sticking-friction axi-symmetric upsetting. It should be a simple matter to change the appropriate entries to tackle different problems. Comment statements are used liberally throughout the program in order to try and explain what is happening at each stage. The programs could have been written in a more condensed and efficient form, but the format given here was chosen deliberately to allow the reader and potential user to follow the program structure and operation as easily as possible. © Copyright 1990 University of Birmingham. Permission to use, copy, modify and distribute this software for any purpose and without fee is hereby granted, provided that the above copyright notice appears in all copies and that both that copyright and this permission notice appear in supporting documentation. The University of Birmingham makes no representations about the suitability of this software for any purpose. The University of Birmingham, the Cambridge University Press and the employees of both organisations disclaim liability for any loss or damage caused through its use.
11 Listing of BASIC program 1000 1010 1020 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390
219
REM=====—================================================================== REM== This is file FILT used to display the on-screen == REM== titles and call the first of the processing programs. == REM== == REM== This is a BASIC finite element program that can be REM== used either for elasticity problems or for elastic == REM== plastic problems based on the small deformation REM== formulation described in chapter 3 and applied in == REM== chapter 4 of the book. The program is intended for == REM== educational or demonstration purposes only. == REM== The program has been developed on an Olivetti M24 == REM== personal computer. It can be used for plane stress, == REM== plane strain or axi-symmetric analysis. == REM== Remember that as a small deformation formulation is == REM== used, the stress components may be unreliable in == REM== the plastic range. == REM=========================================^^ REM CLS PRINT " A Finite Element program for Elasticity or for Elastic-Plastic" PRINT fl problems using a small-deformation formulation" PRINT From the book" PRINT " PRINT PRINT " Finite Element Plasticity and Metalforming Analysis" PRINT PRINT " by" PRINT G.W.Rowe, C.E.N.Sturgess, P.Hartley and I.Pillinger" PRINT " PRINT PRINT " Cambridge University Press, 1990" PRINT PRINT PRINT PRINT PRINT INPUT " press RETURN to continue",A$ REM== Load and run next program == REM CHAIN "FILDATA"
1410 REM== FILT version 1.0 completed 1430 END
16-2-89
Peter Hartley
220
1000
1010 1020 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 1440 1450 1460 1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570 1580 1590 1600 1610 1620 1630
Appendices REM======================== : ================================ = ======= =::::
REM== This is file DISPLAY used to present a graphic display REM== of the finite-element mesh. REM==================—==-=========-===============================:=::== REM REM== Set array dimensions and define variables REM DEFSTR A-B DEFINT E-N DIM CO(25,2),EC(32,3),PO(5) REM REM== Retrieve data from disc file INTLDATA REM OPEN "INTLDATA" FOR INPUT AS 1 INPUT#1,IEP,IGE,NP,NE,NB,NF,INF,NW,INO,INL,IFLAG,IVER INPUT#1,PO(1),PO(2),PO(3) IF IEP=2 THEN INPUT#1/PO(4),PO(5) FOR 1=1 TO NE INPUT#1,EC(I,1),EC(I,2),EC(I,3) NEXT I FOR 1=1 TO NP INPUT#l,CO(Irl),00(1,2) NEXT I CLOSEll REM REM== Shift X and Y coordinates to suit screen display REM FOR 1=1 TO NP CO(I,l)=CO(I,l)+50 CO(I,2)=-CO(I,2)+350 NEXT I REM REM== Set screen mode REM CLS SCREEN 3 LINE (0,0)-C635,0) LINE (635,0)-(635,370) LINE (635,370)-(0,370) LINE (0,370)-(0,0) REM REM== Identify the connecting nodes of each element and REM== plot the element. REM FOR 1=1 TO NE J=EC(I,1) K=EC(I,2) L=EC(I,3) LINE (CO(J,1),CO(J,2))-(CO(K,1),CO(K,2)) LINE (CO(K,1),CO(K,2))-(CO(L,1),CO(L,2)) LINE (CO(L,1),CO(L,2))-(CO(J,1),CO(J,2)) NEXT I REM IF IFLAG=2 THEN GOTO 1690 PRINT PRINT PRINT " Initial finite element mesh" PRINT INPUT " Do you wish to run the FE program? ",A REM REM== return to normal screen mode REM SCREEN 0 IF A="N" THEN GOTO 1810 ELSE IF A="n" THEN GOTO 1810
11 Listing of BASIC program
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221
REM REM== Load next program = REM IF IEP=1 THEN CHAIN "FILBDME" IF INO>1 THEN CHAIN "FILBDMP" ELSE CHAIN "FILBDME" PRINT PRINT INO=INO-1 Finite element mesh at end of increment",INO PRINT " PRINT PRINT IF INO>=INL THEN PRINT " Increment limit has been reached" PRINT INPUT " Do you wish to continue with another increment ",A IF A="N" THEN GOTO 1810 ELSE IF A="n" THEN GOTO 1810 INO=INO+1 GOTO 1610 PRINT PRINT PRINT " ***************** PROGRAM ENDED *****************" REM
1860 REM== DISPLAY version 1.0 completed 20-2-89 Peter Hartley 1870 REM======================================================================== 1880 END
222
1010 1020 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 1440 1450 1460 1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570 1580 1590 1600 1610 1620 1630
Appendices
REM== This is file FILDATA used to create the initial data file. REM== REM== The file has been arranged as a demonstration of the type == REM== of structure required by the finite element program. As == REM== an example the following program has been set up to create == REM== a datafile specifically for simple upsetting with sticking == REM== friction. The data are stored in file INTLDATA. REM== To use this program" for other problems it is necessary == REM== only to change the appropriate entries.If several problems == REM== are to be analysed it would be more useful to use an == REM== interactive version of this program and store the data in == REM== different files. It is advisable to have the mesh details REM== carefully prepared prior to using either type of program. == REM = = = = ==== = = ==== ==: ========== ==: = = = = = ====:===:= = =========: —==================== REM REM REM== Define integer and string variables == REM DEFSTR A-B DEFINT E-N REM REM== Set array dimensions == REM OPTION BASE 1 DIM CO(25,2),LN(10),EC(32,3),NC(15),NT(15),P0(5),R(20),RD(10,2) REM REM REM== Specify whether the problem is elastic or elastic-plastic == REM== IEP=1 for elastic, IEP=2 for elastic-plastic REM IEP=2 REM REM== Specify the geometry of the problem == REM== IGE=1 for axial symmetry REM== IGE=2 for plane stress == REM== IGE=3 for plane strain REM IGE=1 REM REM== Specify the material properties. The elastic modulus in == REM== Newtons per square mm, Poissons ratio and the elastic == REM== yield stress are required for elastic analyses. The == REM== plastic modulus and the plastic Poissons ratio are also == REM== required for elastic-plastic analyses. The values for == REM== each property are stored in array PO respectively. == REM== Linear work hardening is assumed. == REM 1=3 IF IEP=2 THEN 1=5 FOR J=l TO I READ PO(J) NEXT J DATA 50000,0.33,55,100,0.499 REM REM== Specify the number of nodal points in the mesh (NP), == REM== the number of elements (NE), the number of constrained == REM== boundary nodes (NB), the number of boundary nodes with == REM== prescribed finite displacements or with prescribed forces == REM== (NF), and the bandwidth of the stiffness matrix (NW). REM READ NP,NE,NB,NF,NV DATA 9,8,7,3,10 REM
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REM== Specify the nodal connections of each element in an anti REM== clockwise rotation. REM== EC(I) is the list of three nodes defining the element. REM FOR 1=1 TO NE READ EC(I,1),EC(I,2),EC(I,3) NEXT I DATA 1,5,4 DATA 1,2,5 DATA 2,6,5 DATA 2,3,6 DATA 4,8,7 DATA 4,5,8 DATA 5,9,8 DATA 5,6,9 REM REM== Specify the nodal point co-ordinates(mm). x positive is REM== horizontal to the right, y positive is vertical upwards Co-ordinates are stored in the array CO(I,J), I is the REM== nodal point, J=l indicates x, J=2 indicates y. REM== REM FOR 1=1 TO NP READ CO(I,1),CO(I,2) NEXT I DATA 0,0 DATA 50,0 DATA 100,0 DATA 0,50 DATA 50,50 DATA 100,50 DATA 0,100 DATA 50,100 DATA 100,100 REM Specify the constrained boundary nodes and the type of REM== REM== constraint. Nodes are stored in NC(I). REM== Type of constraint is stored in NT(I), NT=1 indicates zero displacement in the y direction, REM== NT=10 indicates zero displacement in the x direction. REM== REM== NT=11 indicates zero displacement in both directions. REM FOR 1=1 TO NB READ NC(I),NT(I) NEXT I DATA 1,11 DATA 2,1 DATA 3,1 DATA 4,10 DATA 7,10 DATA 8,10 DATA 9,10 REM Specify nodes with prescribed displacements or forces REM== REM== and store in LN(I). Specify the x and y components of the displacement(mm) or force(N) and store in RD(I,J) REM==' REM==: J=l indicates x component, J=2 indicates y component. If displacements are being specified set INF to 2, REM== REM== if forces are being specified set INF to 1. If only REM==• one component is being specified then set the other REM== to zero. REM INF=2 REM FOR 1=1 TO NF
223
== ""== ==
== == == ==
== == == == == ==
== == == == == == == ==
224 2280 2290 2300 2310 2320 2330 2340 2350 2360 2370 2380 2390 2400 2410 2420 2430 2440 2450 2460 2470 2480 2490 2500 2510 2520 2530 2540 2550 2560 2570 2580 2590 2600 2610 2620 2630 2640 2650 2660 2670 2680 2690 2700 2710 2720 2730 2740 2750 2760 2770 2780 2790 2800 2810 2820 2830 2840 2850 2860 2870 2880 2890 2900 2910
Appendices READ LN(I),RD(I,1),RD(I,2) NEXT I DATA 7,0,-1 DATA 8,0,-1 DATA 9,0,-1 REM REM== Specify the total number of increments for this analysis == REM== If an elastic treatment has been selected a warning will == REM== be printed if any element will exceed the yield stress. == REM== INO is the increment counter initially set at 1. == REM== INL is the increment limit. REM INO=1 INL=20 REM REM== All the initial data have now been specified. This will == REM== be displayed on the screen prior to storing on disc. == REM CLS IF IEP=1 THEN PRINT "Elastic"; IF IEP=2 THEN PRINT "Elastic-Plastic"; PRINT " Finite Element Analysis in "; IF IGE=1 THEN PRINT "Axial Symmetry" IF IGE=2 THEN PRINT "Plane-Stress" IF IGE=3 THEN PRINT "Plane-Strain" PRINT PRINT "Material Properties" PRINT PRINT " Elastic modulus = ",PO(1) PRINT " Poissons ratio = ",P0(2) PRINT " Yield stress = ",P0(3) IF IEP=2 THEN PRINT " Plastic modulus = ",P0(4) IF IEP=2 THEN PRINT " Plastic ratio = ",PO(5) PRINT PRINT PRINT "The finite element mesh contains "NE" elements, "NP" nodal points," PRINT NB" nodes constrained on the boundaries, and "NF" nodes subject to" PRINT "specified displacements or forces." PRINT "The stiffness matrix has a bandwidth of "NW PRINT INL" Increments have been specified for this analysis." PRINT PRINT PRINT PRINT INPUT " Press RETURN to continue",A CLS PRINT " Element connectivity" PRINT PRINT " Element Nodes defining given element" PRINT FOR 1=1 TO NE PRINT ,I,EC(1,1),ECU,2),EC(1,3) NEXT I INPUT " Press RETURN to continue",A CLS PRINT " Nodal point co-ordinates" PRINT PRINT " Node co-ordinates(mm)" PRINT " x y" FOR 1=1 TO NP PRINT ,1; PRINT USING " ####.##";CO(I,1),C0(I,2) NEXT I INPUT " Press RETURN to continue",A
11 Listing of BASIC program
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2920 CLS 2930 PRINT " Constraints on boundary nodes" 2940 PRINT 2950 FOR 1=1 TO NB 2960 IF NT(I)=10 THEN A="the x direction" 2970 IF NT(I)=1 THEN A="the y direction" 2980 IF NT(I)=11 THEN A="both x and y" 2990 PRINT " Node "NC(D" is constrained in "A 3000 NEXT I 3010 INPUT " Press RETURN to continue",A 3020 CLS 3030 IF INF=2 THEN A=fidisplacements (mm)" 3040 IF INF=1 THEN A="forces (N)" 3050 PRINT " Prescribed nodal point "A 3060 PRINT 3070 PRINT " Node",A 3080 PRINT " x y" 3090 FOR 1=1 TO NF 3100 PRINT ,LN(I),RD(I,1),RD(I,2) 3110 NEXT I 3120 INPUT " Press RETURN to continue",A 3130 PRINT 3140 PRINT " If you wish to see stresses, strains and other results" 3150 PRINT " displayed on the screen at the end of each increment" 3160 PRINT " type Y to the following prompt. If not type N and the" 3170 INPUT " results will^be displayed for the final increment only ",A 3180 IVER=1 3190 IF A="Y" THEN IVER=2 3200 PRINT 3210 PRINT 3220 PRINT "Initial data are now being transferred to disc in INTLDATA" 3230 REM 3240 REM== Open a disc file called INTLDATA in which the initial data 3250 REM== can be stored 3260 REM 3270 OPEN "INTLDATA" FOR OUTPUT AS 1 3280 REM 3290 REM== Store initial data (IFLAG is used in DISPLAY) 3300 REM 3310 IFLAG=1 3320 PRINT#1,IEP;IGE;NP;NE;NB;NF;INF;NV;INO;INL;IFLAG;IVER 3330 PRINT#1,PO(1);PO(2);PO(3) 3340 IF IEP=2 THEN PRINT#1,PO(4);PO(5) 3350 FOR 1=1 TO NE 3360 PRINT#1,EC(I,1);EC(I,2);EC(I,3) 3370 NEXT I 3380 FOR 1=1 TO NP 3390 PRINT#1,CO(I,1);CO(I,2) 3400 NEXT I 3410 FOR 1=1 TO NB 3420 PRINT#1,NC(I);NT(I) 3430 NEXT I 3440 FOR 1=1 TO NF 3450 PRINT#1,LN(I);RD(I,1);RD(I,2) 3460 NEXT I 3470 CLOSEI1 3480 REM 3490 REM== Open a disc file called FORCEV to contain vector of == 3500 REM== specified forces or displacements. == 3510 REM 3520 OPEN "FORCEV" FOR OUTPUT AS 2 3530 REM 3540 REM== Assemble vector. Displacements are multiplied by == 3550 REM== 1E+26 to eliminate any influence from other nodes. ==
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Appendices
REM== The vector is stored in R(I). See also the REM== formulation of the stiffness matrix in FILSTFF. REM FOR J=l TO NP L1=(J-1)*2+1 L2=L1+1 R(L1)=0 R(L2)=0 FOR 1=1 TO NF IF J=LN(I) THEN 3660 ELSE 3680 R(Ll)=RD(I,l)*(INF-l)*lE+26 R(L2)=RD(I,2)*(INF-l)*lE+26 NEXT I PRINT#2,R(L1),R(L2) NEXT J CLOSE#2 REM REM== Disc storage complete REM PRINT PRINT "Data storage is now complete" REM REM== Load and run next program REM PRINT PRINT PRINT INPUT " Do you want a graphical display of the mesh?",A IF A="Y" THEN CHAIN "DISPLAY" IF A="N" GOTO 3870 ELSE INPUT " Please type Y or N",A GOTO 3840 IF IEP=1 THEN CHAIN "FILBDME" IF INO>1 THEN CHAIN "FILBDMP" ELSE CHAIN "FILBDME" REM===================================================================== REM== FILDATA version 1.0 completed 23-1-89 Peter Hartley REM===================================================================== END
11 Listing of BASIC program
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REM== REM==
This is file FILBDME used to set up the [B] and [D] matrices for elasticity problems.
REM REM== Set array dimensions and define variables. REM DEFSTR A DEFINT E-N DEFDBL B-D,O-Z DIM B(4,6),CO(25,2),D(4,4),EC(32,3) ,PO(5) REM REM== Retrieve data from disc file INTLDATA REM OPEN "INTLDATA" FOR INPUT AS 1 INPUT#1,IEP,IGE,NP,NE,NB,NF,INF,NW,INO,INL,IFLAG,IVER INPUT#1,PO(1) ,PO(2) ,PO(3) IF IEP=2 THEN INPUT #1,PO(4) ,PO(5) FOR 1=1 TO NE INPUT#1,EC(I,1),EC(I,2),EC(I,3) NEXT I FOR 1=1 TO NP INPUT#l,CO(Irl),00(1,2) NEXT I CLOSE#1 CLS REM PRINT PRINT PRINT Program FILBDME has been loaded" PRINT PRINT Data are being retrieved from INTLDATA" REM REM== Open files for storing matrices on disc REM OPEN "MATDATB" FOR OUTPUT AS 2 OPEN "MATDATD" FOR OUTPUT AS 3 REM REM REM== Set elastic coefficients for [D] matrix REM T1=(1-PO(2))/(1-2*PO(2)) T2=PO(2)/(1-2*PO(2)) T3=1/(1-PO(2)) T4=PO(2)/(1-PO(2)) TM=PO(1)/(1+PO(2)) REM REM== Call subroutine to set up [B] and [D] matrices REM IF IGE=1 THEN GOSUB 1520 :REM Axial symmetry IF IGE=2 THEN GOSUB 2060 :REM Plane stress IF IGE=3 THEN GOSUB 2500 :REM Plane strain REM REM== Subroutine to set up [D] and [B] matrices for axial symmetry REM REM== [D] Matrix D(1,1)=T1: D(1,2)=T2: D(1,3)=T2: D(l,4)=0 D(2,1)=T2: D(2,2)=T1: D(2,3)=T2: D(2,4)=0 D(3,1)=T2: D(3,2)=T2: D(3,3)=T1: D(3,4)=0 D(4,l)=0: D(4,2)=0: D(4,3)=0: D(4,4)=.5 REM REM== Store matrix in MATDATD FOR 1=1 TO 4 FOR J=l TO 4
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== ==
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Appendices
D(I,J)=D(I,J)*TM PRINT#3rD(I,J) NEXT J NEXT I REM== [B] Matrix REM== Set coefficients and assemble matrix for each element REM FOR N=l TO NE I=EC(N,1) J=EC(N,2) M=EC(N,3) C1=CO(J,1)-CO(I,1) C2=CO(M,1)-CO(I,1) C3=CO(J,2)-CO(I,2) C4=CO(M,2)-CO(I,2) C5=(CO(I,l)+CO(J,l)+CO(M,l))/3 C6=(CO(I,2)+CO(J,2)+CO(M,2))/3 C7=CO(J,1)*CO(M,2)-CO(M,1)*CO(J,2) C8=CO(M,1)*CO(I,2)-CO(I,1)*CO(M,2) C9=CO(I,l)*CO(J,2)-CO(Jfl)*CO(I/2) C10=C7/C5+C3-C4+(C2-C1)*C6/C5 C11=C8/C5+C4-C2*C6/C5 C12=C9/C5-C3+C1*C6/C5 C34=C3-C4 C21=C2-C1 CAREA=(Cl*C4-C2*C3)/2 B(1,1)=C34: B(l,2)=0: B(1,3)=C4: B(l,4)=0: B(l,5)=-C3: B(l,6)=0 B(2/l)=0: B(2,2)=C21: B(2,3)=0: B(2,4)=-C2: B(2/5)=0: B(2,6)=C1 B(3,l)=C10: B(3,2)=0: B(3,3)=C11: B(3r4)=0: B(3,5)=C12: B(3,6)=0 B(4,1)=C21: B(4/2)=C34: B(4/3)=-C2: B(4/4)=C4: B(4,5)=C1: B(4,6)=-C3 REM REM== Store matrix in MATDATB FOR G=l TO 4 FOR H=l TO 6 B(G,H)=B(G,H)/(CAREA*2) PRINT#2,B(G,H) NEXT H NEXT G VOL=CAREA*2*3.14158*C5 PRINT#2,VOL NEXT N RETURN 2940 REM REM== Subroutine to set up [D] and [B] matrices for plane stress REM REM== [D] Matrix D(1,1)=T3: D(1,2)=T4: D(l,3)=0 D(2,1)=T4: D(2,2)=T3: D(2,3)=0 D(3,l)=0: D(3,2)=0: D(3,3)=.5 REM REM== Store matrix in MATDATD FOR 1=1 TO 3 FOR J=l TO 3 D(I,J)=D(I,J)*TM PRINT#3,D(IfJ) NEXT J NEXT I REM== [B] Matrix REM== Set coefficients and assemble matrix for each element REM FOR N=l TO NE I=EC(N,1) J=EC(N,2) M=EC(N,3)
11 Listing of BASIC program
2280 C1=CO(J,1)-CO(I,1) 2290 C2=CO(M,1)-CO(I,1) 2300 C3=CO(J,2)-CO(I,2) 2310 C4=CO(M,2)-CO(I,2) 2320 C34=C3-C4 2330 C21=C2-C1 2340 CAREA=(Cl*C4-C2*C3)/2 2350 B(1,1)=C34: B(l,2)=0: B(1,3)=C4: B(l,4)=0: B(l,5)=-C3: B(l,6)=0 2360 B(2 / l)=0: B<2,2)=C21: B(2,3)=0: B(2,4)=-C2: B(2,5)=0: B(2,6)=C1 2370 B(3,1)=C21: B(3,2)=C34: B(3,3)=-C2: B(3,4)=C4: B(3,5)=C1: B(3,6)=-C3 2380 REH 2390 REM== Store matrix in MATDATB 2400 FOR G=l TO 3 2410 FOR H=l TO 6 2420 B(G,H)=B(G,H)/(CAREA*2) 2430 PRINT#2,B(G,H) 2440 NEXT H 2450 NEXT G 2460 VOL=CAREA 2470 PRINT#2,VOL 2480 NEXT N 2490 RETURN 2940 2500 REM 2510 REM== Subroutine to set up [D] and [B] matrices for plane strain 2520 REM 2530 REM== [D] Matrix 2540 D(1,1)=T1: D(1,2)=T2: D(l,3)=0 2550 D(2,1)=T2: D(2,2)=T1: D(2,3)=0 2560 D(3,l)=0: D<3,2)=0: D(3,3)=.5 2570 REM 2580 REM== Store matrix in MATDATD 2590 FOR 1=1 TO 3 2600 FOR J=l TO 3 2610 D(IfJ)=D(I,J)*TM 2620 PRINT#3,D(I,J) 2630 NEXT J 2640 NEXT I 2650 REM== [B] Matrix 2660 REM== Set coefficients and assemble matrix for each element 2670 REM 2680 FOR N=l TO NE 2690 I=EC(N,1) 2700 J=EC(N,2) 2710 M=EC(N,3) 2720 C1=CO(J,1)-CO(I,1) 2730 C2=CO(M,1)-CO(I,1) 2740 C3=CO(J/2)-CO(I/2) 2750 C4=CO(M/2)-CO(I,2) 2760 C34=C3-C4 2770 C21=C2-C1 2780 CAREA=(Cl*C4-C2*C3)/2 2790 B(1,1)=C34: B(l,2)=0: B(lr 3)=C4: B(l,4)=0: B(l/5)=-C3: B(l,6)=0 2800 B(2,l)=0: B(2,2)=C21: B(2,3)=0: B(2/4)=-C2: B(2/5)=0: B(2/6)=C1 2810 B(3,1)=C21: B(3#2)=C34: B(3,3)=-C2: B(3r4)=C4: B(3,5)=C1: B(3,6)=-C3 2820 REM 2830 REM== Store matrix in MATDATB 2840 FOR G=l TO 3 2850 FOR H=l TO 6 2860 B(G,H)=B(G,H)/
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Appendices NEXT N RETURN 2940 REM REM== Close all disc files REM CLOSE#2 CLOSE#3 REM PRINT PRINT [B] and [D] matrices have been set up and" PRINT " stored in files MATDATB and MATDATD respectively" PRINT " PRINT PRINT The next program is now being loaded" PRINT " REM REM== Load next program REM CHAIN "FILSTFF" REM REM======================== ========== =============== REM= FILBDME version 1.0 completed 20-2-89 Peter Hartley REM=== END
11 Listing of B A S I C program
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1010 REM== This is file FILBDMP used to set up the [B] and [D] 1020 REM== matrices for elastic-plastic problems. == 1030 REM======================================================================== 1040 REM 1050 REM== Set array dimensions and define variables. == 1060 REM 1070 DEFSTR A 1080 DEFINT F-N 1090 DEFDBL B-E,O-Z 1100 DIM B(4,6),CO(25#2)#D(4f4),EC(32,3)fPO(5) 1110 DIM SG(32),SNC(32,3),SNG(32),SNS(32),ST(32,3),STD(32,3),SS(32) 1120 REM 1130 REM== Retrieve data from disc files INTLDATA and STSDATA 1140 REM 1150 OPEN "INTLDATA" FOR INPUT AS 1 1160 INPUT#1,IEP,IGE,NP,NE,NB,NF,INF,NW,INO,INL,IFLAG,IVER 1170 INPUT#1,PO(1),PO(2),PO(3),P0(4),PO(5) 1180 FOR 1=1 TO NE 1190 INPUT#1,EC(I,1),EC(I,2),EC(I,3) 1200 NEXT I 1210 FOR 1=1 TO NP 1220 INPUT#1,CO(I,1),CO(I,2) 1230 NEXT I 1240 CLOSE#1 1250 OPEN "STSDATA" FOR INPUT AS 4 1260 FOR 1=1 TO NE 1270 INPUT#4,STD(I,1),STD(I,2),STD(I,3) 1280 INPUT#4,SG(I),ST(I,1),ST(I,2),ST(I,3),SS(I) 1290 INPUT#4,SNG(I),SNC(I,1),SNC(I,2),SNC(I,3),SNS(I) 1300 NEXT I 1310 CLOSE#4 1320 CLS 1330 REM 1340 PRINT 1350 PRINT 1360 PRINT " Program FILBDMP has been loaded" 1370 PRINT 1380 PRINT " Data are being retrieved from INTLDATA and STSDATA" 1390 REM 1400 REM== Open files for storing matrices on disc == 1410 REM 1420 OPEN "MATDATB" FOR OUTPUT AS 2 1430 OPEN "MATDATD" FOR OUTPUT AS 3 1440 REM 1450 REM== Call subroutine to set up [B] and [D] matrices == 1460 REM 1470 IF IGE=1 THEN GOSUB 1500 :REM Axial symmetry 1480 IF IGE=2 THEN GOSUB 2290 :REM Plane stress :REM Plane strain 1490 IF IGE=3 THEN GOSUB 2890 1500 REM== Subroutine to set up [D] and [B] matrices for axial symmetry == 1510 REM== [D] Matrix 1520 REM== Set coefficients and assemble matrix for each element == 1530 REM 1540 FOR N=l TO NE 1550 REM== Set deviatoric stresses == 1560 SDR=STD(N,1) 1570 SDZ=STD(N,2) 1580 SDT=STD(N,3) 1590 REM== Set flags related to current level of yield stress == 1600 IF SG(N)>PO(3) THEN 1=5 ELSE 1=2 1610 IF SG(N)>PO(3) THEN IM=1 ELSE IM=0 1620 TM=PO(1)/(1+PO(2)) 1630 REM== Evaluate multiplying term for plastic part ==
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Appendices TG=PO(1)/(2*(1+PO(2))) :REM Torsion modulus SD=SG(N)A2*(l+PO(4)/(3*TG))*2/3 SD=1/SD*IM REM== Evaluate coefficients T1=(1-PO(I))/(1-2*PO(I))-SDR*2*SD T2=PO(I)/(1-2*PO(I))-SDR*SDZ*SD T3=PO(I)/(1-2*PO(I))-SDR*SDT*SD T4=-SDR*SS(N)*SD T5=(1-PO(I))/(1-2*PO(I))-SDZ*2*SD T6=PO(I)/(1-2*PO(I))-SDZ*SDT*SD T7=-SDZ*SS(N)*SD T8=(1-PO(I))/(1-2*PO(I))-SDT*2*SD T9=-SDT*SS(N)*SD T10=.5-SS(N)*2*SD REM== Assemble matrix D(1,1)=T1: D(1 / 2)=T2: D(1,3)=T3: D(1,4)=T4 D(2,1)=T2: D(2,2)=T5: D(2,3)=T6: D(2,4)=T7 D(3,1)=T3: D(3,2)=T6: D(3,3)=T8: D(3,4)=T9 D(4,1)=T4: D(4,2)=T7: D(4,3)=T9: D(4,4)=T10 REM== Store matrix in MATDATD FOR 1=1 TO 4 FOR J=l TO 4 D(I,J)=D(I f J)*TM PRINT#3,D(I,J) NEXT J NEXT I NEXT N REM== [B] Matrix REM== Set coefficients and assemble matrix for each element REM FOR N=l TO NE I=EC(N,1) J=EC(N,2) M=EC(N,3) C1=CO(J,1)-CO(I,1) C2=CO(M,1)-CO(I,1) C3=CO(J,2)-CO(I,2) C4=CO(M,2)-CO(I,2) C5=(CO(I / l)+CO(J,l)+CO(M f l))/3 C6=(CO(I # 2)+CO(J,2)+CO(M / 2))/3 C7=CO(J,1)*CO(M,2)-CO(M,1)*CO(J,2) C8=CO(M f l)*CO(I / 2)-CO(I,l)*CO(M,2) C9=CO(I,1)*CO(J,2)-CO(J /1)*CO(I,2) C10=C7/C5+C3-C4+(C2-C1)*C6/C5 C11=C8/C5+C4-C2*C6/C5 C12=C9/C5-C3+C1*C6/C5 C34=C3-C4 C21=C2-C1 CAREA=(Cl*C4-C2*C3)/2 B(l r l)=C34: B(l,2)=0: B(1 / 3)=C4: B(l,4)=0: B(l / 5)=-C3: B(l / 6)=0 B(2,l)=0: B(2 / 2)=C21: B(2,3)=0: B(2,4)=-C2: B(2,5)=0: B(2 f 6)=Cl B(3 f l)"C10: B(3,2)=0: B(3,3)=C11: B(3,4)=0: B(3,5)=C12: B(3 f 6)=0 B(4,1)=C21: B(4,2)=C34: B(4,3)=-C2: B(4,4)=C4: B(4,5)=C1: B(4 / 6)=-C3 REM REM== Store matrix in MATDATB FOR G=l TO 4 FOR H=l TO 6 B(G,H)=B(G,H)/(CAREA*2) PRINT#2,B(G,H) NEXT H NEXT G VOL=CAREA*2*3.14158*C5 PRINT#2,VOL NEXT N
11 Listing of BASIC program
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RETURN 3540 REM== Subroutine to set up [D] and [B] matrices for plane stress REM== [D] Matrix REM== Set coefficients and assemble matrix each element REM FOR N=l TO NE REM== Set deviatoric stresses SDX=STD(N / 1) SDY=STD(N / 1) REM== Set coefficients for elastic-plastic element TP=PO(4)/PO(l)*SG(Nr2*2/9+SS(N) A 2/(l+PO(5)) TR=SDX~2+SDY*2+2*PO(5)*SDX*SDY TQ=2*(1-PO(5)*2)*TP+TR TM=PO(1)/TQ T1=2*TP+SDY~2 T2=2*PO(5)*TP-SDX*SDY T3=-(SDX+SDY*PO(5))*SS(N)/(l+P0(5)) T4=2*TP+SDX*2 T5=-(SDY+SDX*PO(5))*SS(N)/(1+PO(5)) T6=(l-PO(5))*2/9*PO(4)/PO(l)*SG(Nr2+TR/<2*(l+PO(5))) REM== Assemble matrix D(1,1)=T1: D(1,2)=T2: D(1,3)=T3 D(2,1)=T2: D(2,2)=T4: D(2,3)=T5 D(3,1)=T3: D(3,2)=T5: D(3,3)=T6 REM== Store matrix in MATDATD FOR 1=1 TO 3 FOR J=l TO 3 D(I,J)=D(I,J)*TM PRINT#3,D(I,J) NEXT J NEXT I NEXT N REM== [B] Matrix REM== Set coefficients and assemble matrix for each element REM FOR N=l TO NE I=EC(N,1) J=EC(N,2) M=EC(N,3) C1=CO(J,1)-CO(I,1) C2=CO(M,1)-CO(I,1) C3=CO(J,2)-CO(I,2) C4=CO(M / 2)-CO(I f 2) C34=C3-C4 C21=C2-C1 CAREA=(Cl*C4-C2*C3)/2 B(1,1)=C34: B(l f 2)=0: B(1,3)=C4: B(l f 4)=0: B(l,5)=-C3: B(l,6)=0 B(2,l)=0: B(2 / 2)=C21: B(2,3)=0: B(2,4)=-C2: B(2,5)=0: B(2 / 6)=C1 B(3,1)=C21: B(3,2)=C34: B(3,3)=-C2: B(3 / 4)=C4: B(3,5)=C1: B(3 f 6)=-C3 REM REM== Store matrix in MATDATB FOR G=l TO 3 FOR H=l TO 6 B(G / H)=B(G / H)/(CAREA*2) PRINT#2 / B(G,H) NEXT H NEXT G VOL=CAREA PRINT#2,V0L NEXT N RETURN 3540 REM== Subroutine to set up [D] and [B] matrices for plane strain [D] Matrix REM== REM== Set coefficients and assemble matrix for each element
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Appendices
REM FOR N=l TO NE REM== Set deviatoric stresses SDX=STD(N,1) SDY=STD(N,2) REM== Set flags related to current level of yield stress IF SG(N)>PO(3) THEN 1=5 ELSE 1=2 IF SG(N)>PO(3) THEN IM=1 ELSE IM=0 TM=PO(1)/(1+PO(2)) REM== Evaluate multiplying term for plastic part TG=PO(1)/(2*(1+PO(2))) :REM Torsion modulus SD=SG(N)"2*(1+PO(4)/(3*TG))*2/3 SD=1/SD*IM REM== Evaluate coefficients T1=(1-PO(I))/(1-2*PO(I))-SDX*2*SD T2=PO(I)/(1-2*PO(I))-SDX*SDY*SD T3=-SDX*SS(N)*SD T4=(1-PO(I))/(1-2*PO(I))-SDY*2*SD T5=-SDY*SS(N)*SD T6=.5-SS(N)*2*SD REM== Assemble matrix D(1,1)=T1: D(1,2)=T2: D(1,3)=T3 D(2,1)=T2: D(2,2)=T4: D(2,3)=T5 D(3,1)=T3: D(3,2)=T5: D(3,3)=T6 REM REM== Store matrix in MATDATD FOR 1=1 TO 3 FOR J=l TO 3 D(I,J)=D(I,J)*TM PRINT#3,D(I,J) NEXT J NEXT I NEXT N REM== [B] Matrix REM== Set coefficients and assemble matrix for each element REM FOR N=l TO NE I=EC(N,1) J=EC(N,2) M=EC(N,3) C1=CO(J,1)-CO(I,1) C2=CO(M r l)-CO(I,l) C3=CO(J,2)-CO(I,2) C4=CO(M,2)-CO(I,2) C34=C3-C4 C21=C2-C1 CAREA=(Cl*C4-C2*C3)/2 B(1,1)=C34: B(l,2)=0: B(1,3)=C4: B(l,4)=0: B(l,5)=-C3: B(l,6)=0 B(2,l)=0: B(2,2)=C21: B(2,3)=0: B(2,4)=-C2: B(2,5)=0: B(2,6)=C1 B(3,1)=C21: B(3,2)=C34: B(3 r 3)=-C2: B(3,4)=C4: B(3,5)=C1: B(3,6)=-C3 REM REM== Store matrix in MATDATB FOR G=l TO 3 FOR H=l TO 6 B(G / H)=B(G / H)/(CAREA*2) PRINT#2,B(G f H) NEXT H NEXT G VOL=CAREA PRINT#2 / VOL NEXT N RETURN 3540 REM REM== Close all disc files
11 Listing of BASIC program
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REM CLOSE#2 CLOSE#3 REM PRINT PRINT PRINT " [B] and [D] matrices have been set up and" PRINT " stored in files MATDATB and MATDATD respectively" PRINT PRINT PRINT " The next program is now being loaded" REM REM== Load next program REM CHAIN "FILSTFF" REM REM======================================================================== REM==FILBDMP version 1.0 23-1-89 Peter Hartley & Ian Pillinger == REM======================================================================== END
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Appendices
REM== This is file FILSTFF used to assemble the element [k] REM== and global [K] stiffness matrices REM======================================================================= REM REM== Set array dimensions and define variables = REM DEFSTR A DEFINT F-N DEFDBL B-E,O-Z DIM B(4,6),CO(25,2),D(4r4),E(4,6),EC(32,3),LN(10),PO(5),RD(10,2) DIM 8(25,14),SK(6,6) CLS REM REM== Retrieve data from disc file INTLDATA REM OPEN "INTLDATA" FOR INPUT AS 1 INPUTS1,IEP,IGE,NP,NE,NB,NF,INF,NW,INO,INL,IFLAG,IVER INPUT#1,PO(1),PO(2)fPO(3) IF IEP=2 THEN INPUT#1,PO(4),PO(5) FOR 1=1 TO NE INPUT#1,EC(I,1),EC(I,2),EC(I,3) NEXT I FOR 1=1 TO NP INPUT#1,CO(I,1),00(1,2) NEXT I FOR 1=1 TO NB INPUT#1,NC(I),NT(I) NEXT I FOR 1=1 TO NF INPUT#1,LN(I),RD(I,1),RD(I,2) NEXT I CLOSE#1 REM PRINT " FILSTFF has been loaded and the element" PRINT " and nodal point data are being read in " PRINT " from INTLDATA" PRINT PRINT " The global stiffness matrix is now being" PRINT " determined" REM REM== Zero all terms of the banded stiffness matrix REM NEQ=NP*2 :REM 2 is the number of degress of freedom per node FOR 1=1 TO NEQ :REM NEQ is the number of equations for a given problem FOR J=l TO NW :REM NW is the bandwidth of the global stiffness matrix S(I,J)=0 NEXT J NEXT I REM== Open data files REM OPEN "MATDATB" FOR INPUT AS 2 OPEN "MATDATD" FOR INPUT AS 3 REM REM== Consider each element in turn and assemble the global REM== stiffness matrix REM FOR N=l TO NE REM REM== Call subroutine to evaluate element stiffness = REM GOSUB 2220 REM REM== Assemble global matrix
11 Listing of BASIC program
1640 1650 1660 1670 1680 1690 1700 1710 1720 1730 1740 1750 1760 1770 1780 1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2110 2120 2130 2140 2150 2160 2170 2180 2190 2200 2210 2220 2230 2240 2250 2260 2270
237
REM FOR Jl=l TO 3 :REM 3 nodes per element M1=(EC(N,J1)-1)*2 FOR J2=l TO 2 :REM 2 degrees of freedom M1=M1+1 :REM Locates row in global matrix I=(J1-1)*2+J2 :REM Locates row in element matrix FOR J3=l TO 3 M2=(EC(N / J3)-1)*2 FOR J4=l TO 2 L=(J3-1)*2+J4 :REM Locates column in element matrix M3=M2+J4+1-M1 :REM Locates column in banded global matrix IF M3>0 THEN GOTO 1760 ELSE GOTO 1770 S(M1,M3)=S(M1,M3)+SK(I,L) :REM Adds element coefficient to global matrix NEXT J4 NEXT J3 NEXT J2 NEXT Jl NEXT N REM REM== Modify [K] to include zero displacement on specified == REM== boundary nodes == REM FOR N=l TO NB Ml=10 M2=(NC(N)-1)*2 :REM NC(N) is the constrained node FOR M3=l TO 2 M2=M2+1 :REM Locates row in banded matrix I=INT(NT(N)/M1) :REM NT(N) is the type of constraint IF I<=0 THEN GOTO 2010 S(M2,1)=1 :REM Set leading term to unity FOR J=2 TO NW S(M2,J)=0 :REM Set all other terms on specified row to zero M4=M2+1-J IF M4<=0 THEN GOTO 1990 S(M4,J)=0 NEXT J NT(N)=NT(N)-M1*I M1=M1/10 NEXT M3 NEXT N REM REM== Modify [K] to include the influence of specified == REM== node displacements == REM IF INF=1 THEN GOTO 2170 :REM Only required if displacements are specified FOR N=l TO NF M1=(LN(N)-1)*2 FOR J=l TO 2 M1=M1+1 :REM Locates row in banded matrix IF RD(N,J)=0 THEN GOTO 2150 S(Ml,l)=lE+26 :REM Set leading term to large number NEXT J NEXT N REM REM== Close disc files CLOSE#2 CLOSE#3 GOTO 2660 REM REM== Subroutine to evaluate element stiffness REM R E M ™ Input data from disc files REM IF IGE=1 THEN 1=4 ELSE 1=3 :REM axial symmetry or plane stress/strain
238
Appendices
2280 2290 2300 2310 2320 2330 2340 2350 2360 2370 2380 2390 2400 2410 2420 2430 2440 2450 2460 2470 2480 2490 2500 2510 2520 2530 2540 2550 2560 2570 2580 2590 2600 2610 2620 2630 2640 2650 2660 2670 2680 2690 2700 2710 2720 2730 2740 2750 2760 2770 2780 2790 2800 2810 2820 2830 2840 2850 2860 2870
FOR J=l TO I FOR K=l TO 6 INPUT#2,B(J,K) NEXT K NEXT J INPUT!2,VOL IF IEP=1 THEN IF N>1 THEN GOTO 2410 ELSE GOTO 2350 IF IEP=2 THEN IF N>1 THEN IF INO<2 THEN GOTO 2410 ELSE GOTO 2360 FOR J=l TO I FOR K=l TO I INPUT#3,D(J,K) NEXT K NEXT J REM REM== Evaluate [D]*[B] REM FOR J=l TO I FOR K=l TO 6 Z=0 FOR L=l TO I Z=Z+D(J,L)*B(L,K) NEXT L E(J,K)=Z NEXT K NEXT J REM REM== Evaluate [B](transpose)*[D]*[B]*element volume REM FOR J=l TO 6 FOR K=l TO 6 Z=0 FOR L=l TO I Z=Z+B(L,J)*E(L/K) NEXT L SK(J,K)=Z*VOL NEXT K NEXT J RETURN REM REM== Save [K] on disc in file STFDATA REM OPEN "STFDATA" FOR OUTPUT AS 5 FOR 1=1 TO NEQ FOR J=l TO NW PRINT#5rS(I,J) NEXT J NEXT I CLOSEI5 REM PRINT PRINT PRINT " Evaluation of the stiffness matrix is" PRINT " complete" PRINT PRINT " The next program is being loaded" REM REM== Load next program REM CHAIN "FILDSTS" REM
O Q Q A
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'
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==
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i
2890 REM== FILSTFF version 1.0 completed 20-2-89 Peter Hartley 2900 REM=:==:==;:::=================== s=s=============================================== 2910 END
11 Listing of BASIC program 1000 1010 1020 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 1440 1450 1460 1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570 1580 1590 1600 1610 1620 1630
REM=============================================================== REM== This is file FILDSTS used to evaluate nodal point REM== displacements and element stresses REM—============================================================== REM REM== Set array dimensions and define variables REM DEFSTR A DEFINT E-N DEFDBL B-D,O-Z DIM B(4,6),CO(25,2),D(4,4),DEL(6),DIS(20),EC(32,3),LN(10) DIM NC(15),NT(15),PO(5),R(20),RD(10,2) DIM S(25,14),SG(32),SK(6,6),SNC(32,3),SNG(32) DIM SNS(32)/SS(32),ST(32,3),STD(32,3),TI(4)fX(4) REM CLS PRINT " File FILDSTS has been loaded" PRINT PRINT " Displacements, stresses and strains are" PRINT " now being calculated" REM REM== Retrieve data from INTLDATA, FORCEV and STFDATA OPEN "INTLDATA" FOR INPUT AS 1 INPUT#1,IEP,IGE,NP,NE,NB,NF,INF,NW,INO,INL,IFLAG,IVER INPUT#1,PO(1),PO(2),PO(3) IF IEP=2 THEN INPUT#1/PO(4),P0(5) FOR 1=1 TO NE INPUT#1,EC(I,1),EC(I,2) ,EC(I,3) NEXT I FOR 1=1 TO NP INPUTI1,00(1,1),00(1,2) NEXT I FOR 1=1 TO NB INPUT#1,NC(I),NT(I) NEXT I FOR 1=1 TO NF INPUT#1,LN(I),RD(I,1),RD(I,2) NEXT I CLOSE#1 REM OPEN "FORCEV" FOR INPUT AS 2 FOR J=l TO NP L1=(J-1)*2+1 L2=L1+1 INPUT#2,R(L1),R(L2) NEXT J CL0SEI2 REM OPEN "STFDATA" FOR INPUT AS 3 NEQ=2*NP FOR 1=1 TO NEQ :REM NEQ is the number of equations FOR J=l TO NW :REM NW is the bandwidth INPUT|3,S(I,J) NEXT J NEXT I CL0SE#3 REM REM== Temporarily store forces in vector DIS REM FOR 1=1 TO NEQ DIS(I)=R(I) NEXT I REM REM== Reduce equations of global stiffness matrix
239
240
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Appendices
REM FOR N=l TO NEQ I=N FOR L=2 TO NW 1=1+1 IF S(N,L)=0 THEN GOTO 1790 Q=S(N,L)/S(N,1) J=0 FOR K=L TO NW J=J+1 IF S(N,K)=0 THEN GOTO 1760 S(I,J)=S(I,J)-Q*S(N,K) NEXT K S(N,L)=Q DIS(I)=DIS(I)-Q*DIS(N) NEXT L DIS(N)=DIS(N)/S(N,1) NEXT N REM REM== Back substitute to evaluate displacements REM N=NEQ N=N-1 IF N<=0 THEN GOTO 1950 L=N FOR K=2 TO NW L=L+1 IF S(N,K)=0 THEN GOTO 1930 DIS(N)=DIS(N)-S(N,K)*DIS(L) NEXT K GOTO 1860 REM REM== Determine new co-ordinates, store in INTLDATA REM FOR 1=1 TO NP FOR J=l TO 2 L=(I-1)*2+J CO(I/J)=CO(I/J)+DIS(L) NEXT J NEXT I REM== Set flag for graphical display IFLAG=2 REM== Set counter for next increment INO=INO+1 OPEN "INTLDATA" FOR OUTPUT AS 1 PRINT#1,IEP;IGE;NP;NE;NB;NF;INF;NW;INO;INL;IFLAG;IVER PRINT#1,PO(1);PO(2);PO(3) IF IEP=2 THEN PRINT#1,PO(4);PO(5) FOR 1=1 TO NE PRINT#1,EC(I,1);EC(I,2);EC(I,3) NEXT I FOR 1=1 TO NP PRINT#1,CO(I,1);CO(I,2) NEXT I FOR 1=1 TO NB PRINT#l,NCd);NTd) NEXT I FOR 1=1 TO NF PRINT#l,LNd);RDd,D;RDd,2) NEXT I CLOSE#1 REM IF INOMNL THEN GOTO 2280 IF IVER=1 THEN GOTO 2420
11 Listing of BASIC program
2280 2290 2300 2310 2320 2330 2340 2350 2360 2370 2380 2390 2400 2410 2420 2430 2440 2450 2460 2470 2480 2490 2500 2510 2520 2530 2540 2550 2560 2570 2580 2590 2600 2610 2620 2630 2640 2650 2660 2670 2680 2690 2700 2710 2720 2730 2740 2750 2760 2770 2780 2790 2800 2810 2820 2830 2840 2850 2860 2870 2880 2890 2900 2910
CLS PRINT PRINT " PRINT PRINT tf NODE PRINT If HORCO-ORD FOR 1=1 TO NP
241
NODAL DISPLACEMENTS AND NEW CO-ORDINATES" HORDIS VERCO-ORD"
VERDIS";
L2=L1+1 PRINT ,1; PRINT USING NEXT I PRINT INPUT fl Press RETURN to continue",A REM Begin calculation of strains and stresses REM== REM Open disc files REM== REM OPEN "MATDATD" FOR INPUT AS 4 OPEN "MATDATB" FOR INPUT AS 5 ING=INO-1 IF ING>1 THEN OPEN "STSDATA" FOR INPUT AS 6 REM IF IGE=1 THEN 1=4 ELSE 1=3 :REM Axial symmetry or plane stress/strain FOR N=l TO NE :REM NE is the total number of elements REM== If this is the first increment zero strain and stress matrices == IF ING>1 GOTO 2610 STD(N / 1)=O: STD(N,2)=0: STD(N,3)=0 SG(N)=0: ST(N,1)=0: ST(N,2)=0: ST(N,3)=0: SS(N)=0 SNG(N)=0: SNC(N,1)=0: SNC(N,2)=0: SNC(N,3)=0: SNS(N)=0 GOTO 2640 REM== Read data from disc == INPUT#6,STD(N,1),STD(N,2),STD(N,3) INPUT!6,SG(N),ST(N,1),ST(N,2),ST(N,3),SS(N) INPUT#6,SNG(N),SNC(N,1),SNC(N,2),SNC(N,3),SNS(N) FOR J=l TO I FOR K=l TO 6 INPUT#5,B(J,K) :REM Read [B] matrix NEXT K NEXT J INPUT!5,VOL IF IEP=1 THEN IF N>1 THEN GOTO 2780 ELSE GOTO 2710 IF IEP=2 THEN IF N>1 THEN IF ING=1 THEN GOTO 2780 ELSE GOTO 2720 FOR J=l TO I FOR K=l TO I INPUT!4,D(J,K) :REM Read [D] matrix NEXT K NEXT J REM REM== Set up vector of nodal displacements for given element =«= REM FOR Ml=l TO 3 :REM 3 nodes per element FOR M2=l TO 2 :REM 2 degrees of freedom per nodes L1=(EC(N,M1)-1)*2+M2 :REM Location in global vector L2=(M1-1)*2+M2 :REM Location in element vector DEL(L2)=DIS(L1) NEXT M2 NEXT Ml REM REM REM== Evaluate strain components REM FOR J=l TO I
242
2920 2930 2940 2950 2960 2970 2980 2990 3000 3010 3020 3030 3040 3050 3060 3070 3080 3090 3100 3110 3120 3130 3140 3150 3160 3170 3180 3190 3200 3210 3220 3230 3240 3250 3260 3270 3280 3290 3300 3310 3320 3330 3340 3350 3360 3370 3380 3390 3400 3410 3420 3430 3440 3450 3460 3470 3480 3490 3500 3510 3520 3530 3540 3550
Appendices
Z=0 FOR K=l TO 6 Z=Z+B(J,K)*DEL(K) NEXT K X(J)=Z :REM Temporary storage of incremental strains IF J=I THEN SNS(N)=Z ELSE SNC(N,J)=Z NEXT J REM REM== Store stress components and evaluate incremental values REM TI(1)=ST(N / 1) TI(2)=ST(N,2) TI(3)=ST(N,3) TI(4)=SS(N) SG1=SG(N) FOR J=l TO I Z=0 FOR K=l TO I IF K=I THEN Z=Z+D(J,K)*SNS(N) ELSE Z=Z+D(J,K)*SNC(N,K) NEXT K IF J=I THEN SS(N)=Z ELSE ST(N,J)=Z :REM incremental stresses NEXT J REM== Select axial symmetry(1), plane stress(2) or plane strain(3) ON IGE GOTO 3160,3500,3870 REM== GENERALISED STRESS AND STRAIN FOR AXIAL SYMMETRY REM REM== Incremental mean stress and deviatorics SM=(ST(N,l)+ST(N,2)+ST(N,3))/3 TT1=ST(N,1)-SM TT2=ST(N/2)-SM TT3=ST(N,3)-SM REM== Incremental generalised stress based on deviatorics SGEN=SQR(3/2*(TT1*2+TT2*2+TT3'2+2*SS(N)*2)) REM== Incremental generalised strain W=SQR(2/3*(X(1)"2+X(2)*2+X(3)*2+.5*X(4)*2)) REM== Generalised stress based on total strain SNG(N)=SNG(N)+W YSN=PO(3)/PO(1) :REM YSN is the yield stress IF YSN)SNG(N) THEN SG(N)=PO(3)*SNG(N) IF IEP=2 THEN IF YSN<=SNG(N) THEN SG(N)=PO(3)+PO(4)*(SNG(N)-YSN) IF IEP=1 THEN IF YSN<=SNG(N) THEN PRINT "*****VARNING******" IF IEP=1 THEN IF YSN<=SNG(N) THEN PRINT "ELEMENT ",N," HAS YIELDED" REM==Reduce components SRATIO=SG(N)/SGEN IF ING>1 THEN SRATIO=SG(N)/SGI FOR J=l TO 3 ST(N,J)=ST(N,J)+TI(J) ST(N,J)=ST(N,J)*SRATIO IF ING>1 THEN ST(N,J)=TI(J)*SRATIO NEXT J SS(N)=(TI(4)+SS(N))*SRATIO IF ING>1 THEN SS(N)=TI(4)*SRATIO REM Deviatoric stresses HYD=(ST(N,l)+ST(N,2)+ST(N,3))/3 STD(N,1)=ST(N,1)-HYD STD(N,2)=ST(N,2)-HYD STD(N / 3)=ST(N f 3)-HYD GOTO 4230 REM== GENERALISED STRESS AND STRAIN FOR PLANE STRESS REM REM== Incremental mean stress and deviatorics ST(N,3)=0! :REM Axial stress component SM=(ST(N,l)+ST(N,2)+ST(N,3))/3! TT1=ST(N,1)-SM
11 Listing of BASIC program 3560 3570 3580 3590 3600 3610 3620 3630 3640 3650 3660 3670 3680 3690 3700 3710 3720 3730 3740 3750 3760 3770 3780 3790 3800 3810 3820 3830 3840 3850 3860 3870 3880 3890 3900 3910 3920 3930 3940 3950 3960 3970 3980 3990 4000 4010 4020 4030 4040 4050 4060 4070 4080 4090 4100 4110 4120 4130 4140 4150 4160 4170 4180 4190
243
TT2=ST(N/2)-SM TT3=ST(N,3)-SM REM== Incremental generalised stress based on deviatorics SGEN=SQR(3/2*(TT1*2+TT2^2+TT3"2+2*SS(N)"2)) REM== Generalised strain == REM== W is incremental value == REM IF SGEN>PO(3) THEN X(4)=-(X(l)+X(2)) ELSE X(4)=PO(2)/(P0(2)-1)*(X(l)-X(2)) W=SQR(2/3*(X(1)"2+X(2)A2+X<4)*2+.5*X(3)'2)) SNG(N)=SNG(N)+W YSN=PO(3)/PO(1) :REM YSN is the yield strain IF YSN>SNG(N) THEN SG(N)=PO(3)*SNG(N) IF IEP=2 THEN IF YSN<=SNG(N) THEN SG(N)=PO(3)+P0(4)*(SNG(N)-YSN) IF IEP=1 THEN IF YSN<=SNG(N) THEN PRINT "*****WARNING******"; PRINT "ELEMENT ",N," HAS PASSED THE PLASTIC LIMIT" REM== Reduce components SRATIO=SG(N)/SGEN IF ING>1 THEN SRATIO=SG(N)/SGI FOR J=l TO 3 ST(N,J)=ST(N,J)+TI(J) ST(N,J)=ST(N,J)*SRATIO IF ING>1 THEN ST(N,J)=TI(J)*SRATIO NEXT J SS(N)=(TI(4)+SS(N))*SRATIO IF ING>1 THEN SS(N)=TI(4)*SRATIO REM== Deviatoric stresses HYD=(ST(N,l)+ST{N,2)+ST(N,3))/3 STD(N,1)=STD(N,1)-HYD STD(N,2)=STD(N,2)-HYD STD(N,3)=STD(N,3)-HYD GOTO 4230 REM== GENERALISED STRESS AND STRAIN FOR PLANE STRAIN REM REM== Generalised strain REM== W is incremental value REM V=SQR(2/3*(X(1)*2+X(2)*2+.5*X<3)'2)) SNG(N)=SNG(N)+W YSN=PO(3)/PO(1) :REM YSN is the yield strain IF YSN>SNG(N) THEN SG(N)=PO(3)*SNG(N) IF IEP=2 THEN IF YSN<=SNG(N) THEN SG(N)=PO(3)+PO(4)*(SNG(N)-YSN) IF IEP=1 THEN IF YSN<=SNG(N) THEN PRINT "*****WARNING******"; PRINT "ELEMENT f \N," HAS PASSED THE PLASTIC LIMIT" IF YSN)SNG(N) THEN ST(N,3)=PO(2)*(ST(N,1)+ST(N,2)) IF YSN<=SNG(N) THEN ST(N,3)=.5*(ST(N,1)+ST(N,2)) REM== Incremental mean stress and deviatorics == SM=(ST(N,l)+ST(N,2)+ST(N,3))/3! TT^ST^D-SM TT2=ST(N/2)-SM TT3=ST(N/3)-SM REM== Incremental generalised stress based on devaitorics == SGEN=SQR(3/2*(TT1*2+TT2*2+TT3*2+2*SS(N)A2)) REM== Reduce components =: SRATIO=SG(N)/SGEN IF ING>1 THEN SRATIO=SG(N)/SGI FOR J=l TO 3 ST(N,J)=ST(N,J)+TI(J) ST(N/J)=ST(N,J)*SRATIO IF ING>1 THEN ST(N,J)=TI(J)*SRATIO NEXT J SS(N)=(TI(4)+SS(N))*SRATIO IF ING>1 THEN SS(N)=TI(4)*SRATIO REM== Deviatoric stresses HYD=(ST(N,l)+ST(N/2)+ST(N,3))/3
244 4200 4210 4220 4230 4240 4250 4260 4270 4280 4290 4300 4310 4320 4330 4340 4350 4360 4370 4380 4390 4400 4410 4420 4430 4440 4450 4460 4470 4480 4490 4500 4510 4520 4530 4540 4550 4560 4570 4580 4590 4600 4610 4620 4630 4640 4650 4660 4670 4680 4690 4700 4710 4720 4730 4740 4750 4760 4770 4780 4790 4800 4810 4820 4830
Appendices
STD(N#1)=ST(N1rl)-HYD STD(N,2)=ST(N,r2)-HYD STD(N,3)=ST(N,r3)-HYD NEXT N REM== Close files CLOSE#4 CLOSE#5 CLOSE#6 IF I N O M N L THEN GOTO 4300 IF IVER=1 THEN GOTO 4690 REM== Display stresses and strains on screen
REM CLS
PRINT " FOR N=l TO NE PRINT ,N; PRINT USING " NEXT N PRINT INPUT "
ELEMENT GENERALISED STRESS GENERALISED STRAIN"
#######.####";SG(N),SNG(N)
Press RETURN to continue",A
CLS PRINT " A1="XSTRESS": A5="RSTRESS": A9="ELEMENT" IF IGE=1 THEN IF IGE=2 THEN IF IGE=2 THEN PRINT FOR N=l TO NE PRINT N; PRINT USING " NEXT N PRINT INPUT "
STRESS COMPONENTS" A2="YSTRESS": A3="ZSTRESS": A4="XYSTRESS" A6="ZSTRESS": A7="TSTRESS": A8="RZSTRESS" PRINT A9,A5,A6,A7,A8 PRINT A9,A1,A2,A3,A4 PRINT A9,A1,A2,A3,A4
######.###";ST(N,1),ST(N,2),ST(N,3),SS(N)
Press RETURN to continue",A
CLS
PRINT M A1="XSTRAIN": A5="RSTRAIN": IF IGE=1 THEN IF IGE=2 THEN IF IGE=2 THEN PRINT FOR N=l TO NE PRINT N; PRINT USING " NEXT N PRINT INPUT "
STRAIN COMPONENTS" A2="YSTRAIN": A3="ZSTRAIN": A4="XYSTRAIN" A6="ZSTRAIN": A7="TSTRAIN": A8="RZSTRAIN" PRINT A9,A5,A6,A7,A8 PRINT A9,Al,A2 f A3 / A4 PRINT A9,A1 / A2,A3,A4
######.####";SNC(N/1),SNC(N,2),SNC(N,3),SNS(N)
Press RETURN to continue",A
REM
REM==
REM
Store stresses and strains in STSDATA
OPEN "STSDATA'1 FOR OUTPUT AS 7 FOR N=l TO NE PRINT#7.USING'"########.####";STD(N,1);STD(N,2);STD(N,3) PRINT#7,USING1 '########.####";SG(N);ST(N/1);ST(N/2);ST(N,3);SS(N) PRINT#7,SNG(N:>;SNC(N,1);SNC(N,2);SNC(N,3);SNS(N) NEXT N CLOSE#7
REM CLS
PRINT PRINT " PRINT
Increment number ",ING," is complete"
™
11 Listing of BASIC program
484© 4850 4860 4870 4880 4890 4900 4910 4920 4930 4940 4950 4960 4970 4980
245
IF INO>INL THEN GOTO 4860 IF IVER=1 THEN IF IEP=1 THEN CHAIN "FILBDME" ELSE CHAIN "FILBDMP" INPUT " Do you wish to display the distorted mesh",A IF A="Y" THEN CHAIN "DISPLAY" PRINT IF INO>INL THEN PRINT " Specified increment limit has been reached" PRINT INPUT " Do you wish to continue to another increment ",A IF A="N" THEN GOTO 4940 IF IEP=1 THEN CHAIN "FILBDME" ELSE CHAIN "FILBDMP11 PRINT "***********************PROGRAM ENDED**************************" REM======================================================================== Peter Hartley REM== FILDSTS version 1.0 completed 16-2-89 REM======================================================================== END
Bibliography
The bibliography is not set out as a straightforward alphabetical list but has been arranged in groups under the separate headings listed below so that publications on related material are kept together.
Background Reading
Continuum mechanics - tensor theory Finite-element methods Metalworking Numerical modelling of metalforming Numerical techniques Plasticity Research publications
Computer systems Integrated systems Microcomputer applications Finite-element techniques Friction and boundary conditions Re-meshing General finite-element techniques Material behaviour Fracture and defect prediction Forming of new materials (composites, superalloys) Thermo-mechanical analyses Miscellaneous reports Processes Drawing Strip drawing
247 247 247 248 248 248 248 248 249 249 249 250 250 252 253 256 256 257 259 261 266 266 266
Bibliography
Wire drawing Extrusion Backward extrusion Combined extrusion Forward extrusion Forging Axi-symmetric forging Flat tool forging Plane-strain forging Three-dimensional forging Upsetting Heading Indentation Rolling Plane-strain rolling Shape rolling Three-dimensional rolling Review papers Theoretical background Finite-element theory Elastic-plastic finite-element theory Rigid-plastic finite-element theory Visco-plastic finite-element theory Plasticity theory Unpublished theses
247
266 266 266 267 267 270 270 272 272 273 274 277 277 278 278 280 280 281 282 282 282 287 287 288 289
BACKGROUND READING The publications in this section will be particularly useful for anyone approaching the subject of finite-element simulation of metalforming for the first time.
Continuum mechanics - tensor theory Eringen, A.C. Nonlinear Theory of Continuous Media, McGraw-Hill (1962). Hunter, S.C. Mechanics of Continuous Media, Ellis Horwood (1976). Leigh, D.C. Nonlinear Continuum Mechanics, McGraw-Hill (1968). Spencer, A.J.M. Continuum Mechanics, Longman (1980).
Finite-element methods Cheung, Y.K. and Yeo, M.E A Practical Introduction to Finite Element Analysis, Pitman (1979).
248
Bibliography
Hinton, E. and Owen, D.R.J. Finite Element Programming, Academic Press (1977). Irons, B. and Ahmad, S. Techniques of Finite Elements, Halsted Press (1980). Livesley, R. Finite-elements: An Introduction for Engineers, Cambridge University Press (1984). Rao, S.S. The Finite Element Method in Engineering, Pergamon Press (1982). Zienkiewicz, O.C. The Finite-Element Method, McGraw-Hill (1977).
Metalworking Avitzur, B. Metalforming Processes and Analysis, McGraw-Hill (1968). Rowe, G.W. Principles of Industrial Metalworking Processes, Arnold (1977).
Numerical modelling of metalforming Boer, C.R., Rebelo, N., Rydstad, H. and Schroder, G. (Eds.) Process Modelling of Metal Forming and Thermomechanical Treatment, Springer (1986). Owen, D.R.J. and Hinton, E. Finite Elements in Plasticity: Theory and Practice, Pineridge Press (1980). Pittman, J.F.T., Zienkiewicz, O . C , Wood, R.D. and Alexander, J.M. (Eds.) Numerical Analysis of Forming Processes, Wiley (1984).
Numerical Techniques Boas, M.L. Mathematical Methods in the Physical Sciences, Wiley (1983). Day, A.C. Fortran Techniques, Cambridge University Press (1972). Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T. Numerical Recipes, Cambridge University Press (1986).
Plasticity Ford, H. and Alexander, J.M. Advanced Mechanics of Materials, Ellis Horwood (1977). Hill, R. The Mathematical Theory of Plasticity, Clarendon Press (1950). Hoffmann, O. and Sachs, G. Introduction to the Theory of Plasticity for Engineers, McGraw-Hill (1953). Johnson, W. and Mellor, F.B. Engineering Plasticity, von Nostrand, Reinhold (1973). Nadai, A. Plasticity, a Mechanics of the Plastic State of Matter, McGraw-Hill (1931). Nadai, A. Theory of Flow and Fracture, McGraw-Hill (1950). Tabor, D. The Hardness of Metals, Oxford University Press (1951).
RESEARCH PUBLICATIONS This section is intended to represent a comprehensive and up-to-date survey of publications relating to the finite-element simulation of metalforming. The
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authors apologise in advance for any omissions or errors, and will be happy to receive suggestions for inclusion in any future editions of this monograph.
Computer systems Integrated systems
Altan, T. and Oh, S.I. CAD/CAM of tooling and process for plastic working. Proc. 1st Int. Conf. on Technology of Plasticity, ed. H. Kudo, JSTP/JSPE, pp. 531-44 (1984). Eames, A.J., Dean,T.A., Hartley, P. and Sturgess, C.E.N. An integrated computer system for forging die design and flow simulation. Proc. Int. Conf. on Computer Aided Production Engineering, ed. J.A. McGeough, Mechanical Engineering Pubs., pp. 231-6 (1986). Eames, A.J., Dean, T.A., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. An IKBS for upset forging die design. Proc. 2nd Int. Conf. on Computer Aided Production Engineering, ed. J.A. McGeough, Mechanical Engineering Pubs., pp. 37-41 (1987). Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Expert systems for design and manufacture. Proc. 12th All India Machine Tool Des. Res. Conf.,ed. U.R.K. Rao, P.N. Rao and N.K. Tiwari,Tata McGraw-Hill, pp. xxi-xxvii (1986). Hartley, P., Sturgess, C.E.N., Dean,T.A. and Rowe, G.W. Forging die design and flow simulation: their integration in intelligent knowledge based systems. /. Mech. Wkg. Techn. 15, 1-13 (1987). Hartley, P., Sturgess, C.E.N., Dean,T. A., Rowe, G.W. and Eames, A.J. Development of a forging expert system Expert Systems in Engineering, ed. D.T. Pham, IFS Pubs., pp. 425-43 (1988). Kopp, R. and Arfmann, G. The application of CAD/CAE/CAM from the viewpoint of plastic working technology. Proc. 1st Int. Conf on Technology of Plasticity, ed. H. Kudo, JSTP/JSPE, pp. 489-97 (1984). Pillinger, I., Hartley, P., Sturgess, C.E.N. and Dean,T.A. An intelligent knowledge based system for the design of forging dies. Artificial Intelligence in Design, ed. D.T Pham, IFS Pubs. (1990). Pillinger, I., Sims, P., Hartley, P., Sturgess, C.E.N. and Dean,T.A. Integrating CAD/CAM of forging dies with finite-element simulation of metal flow in an intelligent knowledge based system. Integrating Information and Material Flow, Proc. Int. Conf. on Factory 2000, Institution of Electronic and Radio Engineers, pp. 205-11 (1988). Microcomputer applications
Dung, N.L. The use of personal computer for FE simulation of forging process. Proc. 11th CANCAM Conf., Edmonton, Alberta (1987). Fadzil, M., Chuah, K.C., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Metalforming analysis on desk-top microcomputers using non-linear elasticplastic finite element techniques. Proc. 22nd Int. Machine Tool Des. Res. Conf., ed. B.J. Davies, Macmillan, pp. 533-9 (1981). Hussin, A.A.M., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Finite element plasticity on microcomputers. Proc. Stress Analysis and the Micro Conf. (SAM'85), City University, London, pp. 106-15 (1985). Hussin, A.A.M., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Non-linear
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finite-element simulation of metalforming processes on a 16 bit microcomputer. Proc. 2nd Int. Conf. on Microcomputers in Engineering: Development and Application of Software, ed. B.A. Schrefler and R.W. Lewis, Pineridge Press, pp. 517-28 (1986). Hussin, A.A.M., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Non-Linear finite-element analysis on microcomputers for metal forging. /. Strain Analysis 21, 197-203 (1986). Hussin, A.A.M., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Simulation of cold industrial forming processes. Special issue of /. Comm. Appl. Num. Meth. 3, 415-26 (1987). Hussin, A.A.M., Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Elastic-plastic finite-element modelling of cold backward extrusion using a microcomputer based system. / . Mech. Wkg. Techn. 16, 7-20 (1988). Rowe, G.W. Finite element analysis in metalforming. Ill Jugoslav, Simpozijum o Metalurgiji, Beograd, pp. 53-65 (1984).
Finite-element techniques Friction and boundary conditions Baaijens, F.P.T., Veldpaus, F.E. and Brekelmans, W.A.M. On the numerical simulation of contact problems in forming processes. Proc. 2nd Int. Conf. on Numerical Methods in Industrial Forming Processes, ed. K. Mattiasson, A. Samuelsson, R.D. Wood and O.C. Zienkiewicz, Balkema Press, pp. 85-90 (1986). Charlier, R. and Habraken, A.M. On the modelling of three dimensional contact with friction problem in context of large displacement problems. Supplement to Proc. 2nd Int. Conf. on Numerical Methods in Industrial Forming Processes, ed. K. Mattiasson, A. Samuelsson, R.D. Wood and O.C. Zienkiewicz, Balkema Press, (1986). Charlier, R., Godinas, A. and Cescotto, S. On the modelling of contact problems with friction by the finite element method. Proc. 8th Conf. on Structural Mechanics in Reactor Technology, Brussels, (1985). Chen, C.C. and Kobayashi, S. Rigid plastic finite element analysis of ring compression. Applications of Numerical Methods to Forming Processes, Winter Annual Meeting of ASME, AMD 28, ed. H. Armen and R.F. Jones Jr., ASME, pp. 163-74 (1978). Gunasekera, J.S. and Mahadeva, S. An alternative method to predict the effects of friction in metalforming. Friction and Material Characterizations, Winter Annual Meeting of ASME, MD 10, ed. I. Haque; J.E. Jackson Jr., A. A. Tseng and J.L. Rose, ASME, pp. 55-62 (1988). Haber, R.B. and Hariandia, B.H. An Eulerian-Lagrangian finite element approach to large-deformation frictional contact. Comput. Struct. 20, 193201 (1985). Hallquist, J.O., Goudreau, G.L. and Benson, D J . Sliding interfaces with contact-impact in large-scale Lagrangian computations. Comp. Meth. Appl. Mech. Eng. 51, 107-37 (1985). Hartley, P., Pillinger, I. and Sturgess, C.E.N. Modelling frictional effects in finite-element simulation of metal forming. Friction and Material Characterizations, Winter Annual Meeting of ASME, MD 10, ed. I. Haque, J. E. Jackson Jr., A.A. Tseng and J.L. Rose, ASME, pp. 91-6 (1988).
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Hartley, P., Sturgess, C.E.N. and Rowe, G.W. An examination of frictional boundary conditions and their effect in an elastic-plastic finite element solution. Proc. 20th Int. Machine Tool Des. Res. Conf., ed. S.A.Tobias, Macmillan, pp. 157-63 (1979). Hartley, P., Sturgess, C.E.N. and Rowe, G.W. Friction in finite-element analyses of metalforming processes. Int. J. Mech. Sci. 21, 301-11 (1979). Hartley, P., Sturgess, C.E.N. and Rowe, G.W. A prediction of the influence of friction in the ring test by the finite element method. Proc. 7th Nth. American Metalworking Res. Conf., SME, pp. 151-8 (1979). Huetink, J., Van der Lugt, J. and Miedama, J.R. A mixed Eulerian-Lagrangian contact element to describe boundary and interface behaviour in forming processes. Proc. Int. Conf. on Numerical Methods in Engineering, Theory and Applications (NUMETA), Martinus-Nijhof (1987). Hwang, S.M. and Kobayashi, S. A note on evaluation of interface friction in ring tests. Proc. 11th Nth. American Metalworking Res. Conf., SME, pp. 193-6 (1983). Jackson Jr., J.E. Haque, I., Gangjee,T. and Ramesh, M. Some numerical aspects of frictional modeling in material forming processes. Friction and Material Characterizations, Winter Annual Meeting of ASME, MD 10, ed. I. Haque, J.E. Jackson Jr., A.A. Tseng and J.L. Rose, ASME, pp 39-46 (1988). Kikuchi, N. and Cheng, J-H. Finite element analysis of large deformation problems including unilateral contact and friction. Comp. Meth. for Non-linear Solids and Structural Mechanics, AMD 54, ed. S.N. Alturi, ASME pp.
121-32 (1983). Kobayashi, S. Thermoviscoplastic analysis of metal forming problems by the finite element method. Proc. 1st Int. Conf on Numerical Methods in Industrial Forming Processes, ed. J.F.T. Pittman, R.D. Wood, J.M. Alexander and O.C. Zienkiewicz, Pineridge Press, pp. 17-25 (1982). Kopp, R. and Cho, M.L. Influence of the boundary conditions on results of the finite-element simulation. Proc. 2nd Int. Conf on Technology of Plasticity, ed. K. Lange, Springer, pp. 43-50 (1987). Mahrenholtz, O. Different finite element approaches to large plastic deformations. Comp. Meth. Appl. Mech. Eng. 33, 453-68 (1982). Makinouchi, A., Ike, H., Murakawa, M., Noga, N. and Ciupik, L.F. Finite element analysis of flattening of surface asperities by rigid dies in metal working processes. Proc. 2nd Int. Conf on Technology of Plasticity, ed. K. Lange, Springer, pp. 59-66 (1987). Matsumoto, H., Oh, S.I. and Kobayashi, S. A note on the matrix method for rigid-plastic finite element analysis of ring compression. Proc. 18th Int. Machine Tool Des. Res. Conf., ed. J.M.Alexander, Macmillan, pp. 3-9 (1977). Mori, K., Osakada, K., Nakadoi, K. and Fukuda, M. Coupled analysis of steady state forming process with elastic tools. Proc. 2nd Int. Conf on Numerical Methods in Industrial Forming Processes, ed. K. Mattiasson, A. Samuelsson, R.D. Wood and O.C. Zienkiewicz, Balkema Press, pp. 237-42 (1986). Oh, S.I. Finite element analysis of metal forming processes with arbitrarily shaped dies. Int. J. Mech. Sci. 24, no. 8, 479-93 (1982). Oh, S.I., Rebelo, N. and Kobayashi, S. Finite element formulation for the analysis of plastic deformation of rate sensitive materials in metal forming. Metal Forming Plasticity, ed. H. Lippmann, Springer, pp. 273-91 (1979). Pillinger, I., Hartley, P. and Sturgess, C.E.N. Modelling of frictional tool surfaces
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Review papers Cheng, J-H. and Kikuchi, N. An analysis of metal forming processes using large deformation elastic-plastic formulations. Comp. Meth. Appl. Mech. Eng. 49, 71-108 (1985). Haque, I., Jackson Jr., J.E., Gangjee, T. and Raikar, T. Empirical and finite element approaches to forging die design: a state-of-the-art survey. /. Mater. Shaping Techn. 5, 23-33 (1987). Jimma,T. ,Tomita, Y. and Shimamura, S. Benchmark test on a plastic deformation problem, application of numerical methods of analysis to the uniaxial tension of a block or cylindrical bar with both ends fixed. Proc. 2nd Int. Conf. on Technology of Plasticity, ed. K. Lange, Springer, pp. 73-80 (1987).
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Index
anisotropy 175 assembly of element stiffness matrices 10, 90 automobile spigot 159 axial symmetry 24 axi-symmetric upsetting 52, 58, 63 [B] matrix 20 derivation of 178 boundary conditions 6, 10, 35, 83 imposition of 196 Boundary-Element Method 3 boundary surfaces displacement of 86 rotation of 88 bulk modulus, elastic 76, 203 calculus of variations 68 Cauchy stress 100, 212 reference components of 211 Cartesian co-ordinates 67 catastrophic shear 2 cavity formation 171 Choleski decomposition, see solution methods for matrix equations compiler limitations in FORTRAN 61 composites 171 compression of cube 99 Computer-Aided Design (CAD) 2 Computer-Aided Manufacturing (CAM) 2 computer hardware selection 49, 52 computer program transfer from mainframe to micro 58 computer software selection 49 conductivity at die interface 138 constitutive equation 41, 45 constraining conditions, nodal 83 contact between surfaces 85 continuity between finite elements 18 contravariant components 205 convected co-ordinate system 209 convergence of a finite-element solution 19 co-processor for a microcomputer 62 covariant components 206
crack propagation 171 cracking centre, in plane-strain side-pressing 131 corner, in plane-strain side-pressing 131 tensile 2 [D] matrix 22 derivation of elastic 181 derivation of elastic-plastic 184 elastic 187 plastic 187 data input 87 deformation gradient 71 increment 71 mapping into orthogonal and symmetric components 72 matrix 213 prescribing values of 85 rate tensor 67, 68, 217 stiffness matrix 78 degrees of freedom for a node 75 development lead times 116 deviatoric strain increment 184 deviatoric stress 42, 96 spatial gradients of 99 die filling 116 die shape 80 die stressing 116 die surface, modelling 83 dilatation correction matrix 78 discretisation 6, 14 displacement components of, prescribed in global axis system 196 components of, prescribed in rotated axis system 198 increment of nodal 77 of boundary surfaces 86 of surface nodes 8 vector 6, 20 vector, global 9 ductile fracture 116, 127 dyadic product of tensors 67
Index elastic bulk modulus 76 elastic deformation 2 of a single cube 69 elastic finite-element analysis 14 elastic hysteresis loss 8 elastic modulus of rigidity 22 elastic-plastic theory constitutive equation 45, 74 constitutive matrix 7 derivation of [D] matrices for 181-7 small-deformation 40 finite-strain 66 flow rule 74, 184 element dilatation 75 element stiffness matrix, assembly 10 energy change in, due to heat conduction 101, 105 due to frictional work 101 due to work of deformation 101 expended in forging 3 minimisation of rate of potential 68 potential 7, 25, 188 principle of minimum (Hamiltonian principie) 7 principle of minimum potential 4, 9, 25 rate of increase of internal deformation 67 rate of increase of potential 67 shear strain 41 strain 5, 8, 188 equilibrium equations 6 of forces at a node 18 expert system 81, 174 external forces 100 extrusion backward 133, 165 forward 149 extrusion-forging 135 fatigue resistance 2 finite-difference method 3, 38 finite-element analysis basic formulation for elastic deformation 14 general procedure for structures 5 using BASIC 51 using FORTRAN on a microcomputer 52 finite-elements aspect ratio of 18 constant strain triangle 21, 188 continuity between 18 compatibility of strains between 18 higher order (quadratic) 16 linear 15 linear quadrilateral 16 pin-jointed bar 4 plane-strain triangular 16 size of 17 three-dimensional 17 triangular plate 4 types of 14 finite-strain formulation 66 implementation of 80-115 flow divide
295
in extrusion-forging 136 in rolling 154 flow stress 41 flow vectors 141 force equilibrium of nodal 18 vector 6 forging extrusion- 135 of a connecting rod 145-9 of a rectangular block 117 plane-strain ('H'-section) 137 rim-disc 131 sequence design for 159 upset 122 warm 138 fracture, prediction of 127-31 fracture criterion, generalised plastic work 130 friction factor 91 friction in FE analyses 90, 102 friction layer 91 frontal-solution technique 112 Galerkin's method 38 Gaussian elimination see solution methods for matrix equations Gaussian quadrature 78 global axis system 83 gudgeon pin, forming of 163 Hamiltonian principle, see principle of minimum energy Hardness, Vickers Pyramid Number 121, 126, 134, 152, 162 heading 60, 63, 122 heat conduction 102 hydrostatic stress 42, 72, 98 incompressibility, plastic 74 incremental displacement of a particle 77 incremental element-stiffness equations 78 incremental equations 70 incremental rotations 86 initial mesh geometry 86 initial state parameters 86 initial-stiffness solution technique 92 interpolation function for field variable in a continuum 6 for nodal displacements 77, 100 polynomials 19 interpolation during re-meshing 109 isothermal plastic deformation 40 Jacobian 78 of deformation 211 Kirchhoff stress, see stress Kronecker delta 42, 69, 182, 206 Lagrange stress, see stress Levy-Mises flow rule 184 load vector 6 material properties 85, 172
296
Index
matrix equations, see solution methods for matrix equations mean-normal method 96 mechanical press 86 mesh generation 82 metalforming operation 81 microcomputer applications 49-65 minimum potential energy 9, 25 modulus of elasticity 22 modulus of plasticity 69 modulus of rigidity 202 multi-stage processes 116, 159, 163 neutral point in extrusion-forging 136 in rolling 154 Newtonian fluid 44 Newton's method of iteration 96 node constraining conditions 83 displacement, interpolation function 77 displacement vector 6 nominal stress 67, see also stress operating system on a microcomputer 57 operational parameters 116 orthonormal basis 206 orthonormal Cartesian system 217 orthonormal matrix 199 output of FE results 106 paging 133 pin-jointed bar element 4 Piola-Kirchhoff stress, see stress plane-strain forging 137 plane-strain side-pressing 125 plastic deformation constancy of volume in 42 isothermal 40 large-strain 66 von Mises yield in 41, 74 yielding in 40, 73 plastic incompressibility 74 plastic modulus 69 rate of change of 185 Poisson's Ratio 22, 203 polymer processing 175 post-processing 80, 108, 173 potential energy 7, 25 rate of increase of 67 minimisation of rate of 68 practical applications of the FE technique 11668 Prandtl-Reuss equations 44, 74, 184 pre-processing 82 pressure distribution in rolling 156 primitive geometric surfaces 84 principle of minimum energy 7 product properties 116 rate equations, governing 67 Rayleigh-Ritz method 37 re-meshing 109, 133
residual stress 116 rigid-body rotation 68 rigidity, elastic modulus of 22, 182, 202 rim-disc forging 131 ring-test 91 rolling 110 friction boundary conditions in 153 slab 155 strip 153 rotated set of axes 83 rotation, effect of 71 rotation of boundary surfaces 88 rotation of stressed element 98 rotational matrices 72 rotational transformation matrix 199 secant-modulus method 94 severe deformation 108 shape function, Cartesian spatial derivatives 77 shear catastrophic 2 strain energy 41 strain in 2-D 22 stress 22 yield 90 sheet forming 175 slab rolling 155 solution methods for matrix equations 31 Choleski LU decomposition 32 conjugate-gradient method 34 direct solution methods 31 frontal-solution technique 112 Gaussian elimination and back-substitution 31, 191 Gauss-Jordan elimination 31 incomplete Choleski conjugate-gradient method 34 indirect solution methods 33 initial stiffness technique 92 iteration to improve a direct solution 35 secant-modulus 94 Runge-Kutta, second-order 94 spread in slab rolling 158 steady-state processes 111 stiffness, incremental element equations 78 stiffness matrix assembly and solution 90, see also solution methods for matrix equations bandedness of 28 deformation stiffness matrix 78 derivation of, for plane-stress triangular element 188 element 4 for small deformation 46 formulation of element 25 formulation of global 26 global 191 influence coefficient in 15 positive semi-definiteness of 27 properties of 27 sparseness of diagonal band of 29, 195 symmetry of 27
Index
strain bulk (volume) 76 bulk, increment 76 compatibility between elements 18 decomposition of incremental 43 dilatational element of 42 distortional element of 42 elastic, in a tensile test 12 energy 5 energy, shear 41 engineering, in a tensile test 4 engineering shear 16 finite, formulation 66 generalised 43 generalised plastic increment of 43 hardening in an FE solution 45, 96 increment, deviatoric 75, 184 increment, normality to yield locus 74, 184 increment, plastic 97, 184 increment tensor 95 increment vector 187 infinitesimal definition of 72 linearised co-rotational, increment of 72,77 logarithmic 40 plane 16 plane elastic 24 rate 68, 100 rate tensor 69 shear, in 2-D 22 strain-displacement [B] matrix 20 stress Cauchy (true) 98, 100, 213 Cauchy (true), rate of 68 deviatoric 42, 96 elastic, in a tensile test 12 flow 41 generalised 42 hydrostatic 42, 72, 99 in a deforming body 209 increment correction matrix 78 increment vector 187 Jaumann derivative of Kirchhoff 69, 216 Kirchhoff (Piola-Kirchhoff II) 212, 214 Kirchhoff, rate of 68 Lagrange (Piola-Kirchhoff I) 211 nominal 67, 211, 214 nominal, rate of 69, 211 plane, elastic 23 rate 69 rates 216-17 reference components of Cauchy 211 shear 22
297
shear yield 90 spatial gradients of deviatoric 99 tensor, average 67 yield, in pure shear 41 stress-strain [D] matrix 22 strip rolling 153 subscript notation 182 suffix summation convention 182, 205 super-grid technique (re-meshing) 111, 133,151 temperature 101 tensile cracking 2 tensile test 4 elastic analysis of 7 elastic-plastic analysis of 46 tensors 205-8 thermal capacity 101, 138 thermal conductivity 138 thermal step 101 three-dimensional analyses 112 tool production, sequences for 2 triangular plate element 4 unit vector 210 upper-bound technique 3 upset forging 122 upsetting, axi-symmetric 52, 58, 63 variational methods of solution in continuum theory 36 variational principle 4 vectors 205-8 velocity field 68 virtual memory 113 volume constraint 76 von Mises yield criterion 41, 74, 184 warm forging 138 wear 176 weighted-residual method 37 work done by applied forces 188 in a tensile test 5, 8 work of deformation 101 workpiece geometry 80 yield criterion 73, 184 yield locus, initial 96 yield stress in pure shear 41 yield transition increment 95 Young's Modulus of Elasticity 4